Is the reason that Lysa’s death was not particularly investigated in depth by the Lords of the Vale because they were relieved to be rid of her instability(and eager to try and seize control of Robert Arryn)?

Those were some of the reasons. There were others:

First, there was no physical evidence and very few witnesses, who all told the same story. So even if they wanted to do an in-depth investigation, there wasn’t much to go on.
Second, even if they had wanted to, I’m pretty sure that Nestor Royce, as High Steward and Keeper of the Gates of the Moon, would have jurisdiction over the crime. And since Littlefinger had bought him off…
Third, Marillion had confessed to the crime, and Marrillion was disliked for his arrogance and for singing satirical songs about various noblemen who came courting Lysa.

I hope si not a stupid question, but why did the vale lords want to join the starks in the wo5k? What do they do when robb is crowned ? Thank you

Not a stupid question at all.

There were substantial numbers of Vale lords who wanted to fight the Lannisters for several reasons:

“Lord Jon was much loved, and the insult was keenly felt when the king named Jaime Lannister to an office the Arryns had held for near three hundred years. Lysa has commanded us to call her son the True Warden of the East, but no one is fooled. Nor is your sister alone in wondering at the manner of the Hand’s death. None dare say Jon was murdered, not openly, but suspicion casts a long shadow.” (Catelyn VI, AGOT)

“Pycelle pushed himself to his feet. He was clad in a magnificent robe of thick red velvet, with an ermine collar and shiny gold fastenings. From a drooping sleeve, heavy with gilded scrollwork, he drew a parchment, unrolled it, and began to read a long list of names, commanding each in the name of king and council to present themselves and swear their fealty to Joffrey. Failing that, they would be adjudged traitors, their lands and titles forfeit to the throne.

The names he read made Sansa hold her breath. Lord Stannis Baratheon, his lady wife, his daughter. Lord Renly Baratheon. Both Lord Royces and their sons.” (Sansa V, AGOT)

“Lysa was as lonely as she was. Her new husband seemed to spend more time at the foot of the mountain than he did atop it. He was gone now, had been gone the past four days, meeting with the Corbrays. From bits and pieces of overheard conversations Sansa knew that Jon Arryn’s bannermen resented Lysa’s marriage and begrudged Petyr his authority as Lord Protector of the Vale. The senior branch of House Royce was close to open revolt over her aunt’s failure to aid Robb in his war, and the Waynwoods, Redforts, Belmores, and Templetons were giving them every support.” (Sansa VII, ASOS)

The Lords of the Vale don’t declare for Robb Stark because Lysa has summoned her knights to the Eyrie and refuses to let them leave – “If it were up to me, I would take a hundred men into the mountains, root them out of their fastnesses, and teach them some sharp lessons, but your sister has forbidden it. She would not even permit her knights to fight in the Hand’s tourney. She wants all our swords kept close to home, to defend the Vale” – although given her close cooperation with Petyr Baelish later in ASOS, we can surmise that the reason why the knights of the Vale were kept at home is that A. Littlefinger didn’t want Robb Stark to win the War of Five Kings, and B. he wanted to keep them fresh for his attempt to win the Iron Throne.

“Timett son of Timett…he may have Arryn blood in him…” Whoa wait what? How did I miss this?

So let’s talk about Timett son of Timett.

The Burned Men are a particularly fearsome mountain clan from the Vale, who broke off from the Painted Dogs to worship a fire-witch (almost certainly the fugitive Nettles with her dragon Steepstealer). The Burned Men are known for their fearlessness demonstrated in their coming-of-age ceremony where they burn off a body-part, a ceremony deriving from the practice of sending young men as tribute to the fire-witch who would have to prove themselves by daring the flames of Sheepstealer. And the Burned Men once carried off the daughter of Alys Arryn (Jon Arryn’s sister)…

Timett became a red hand (war chief) of the Burned Men from a very early age by burning out one of his eyes at the coming-of-age ceremony, which scared the hell out of the normally unflappable Burned Men. Timett fought for Tyrion at the Green Fork and Blackwater with some distinction, and then returned to the Vale with the spoils of war:

“The Burned Men are fearless since Timett One-Eye came back from the war. And half a year ago, Gunthor son of Gurn led the Stone Crows down on a village not eight miles from here. They took every woman and every scrap of grain, and killed half the men. They have steel now, good swords and mail hauberks, and they watch the high road—the Stone Crows, the Milk Snakes, the Sons of the Mist, all of them. Might be you’d take a few with you, but in the end they’d kill you and make off with your daughter.“

So now you have the mountain clans armed and armored equivalent to knights, with experience in fighting knights in open battle, and Timett leads the toughest of their clan.

My speculation is that, once the Knights of the Vale go north to pursue Sansa’s claim to Winterfell, Timett will seize the Eyrie. And in a bit of dramatic irony, it’ll turn out that he’s actually the rightful heir, as his claim outranks that of Harry the Heir (who descends from the youngest of Alys Arryn’s daughters).

What exactly is the geopolitical significance of the Three Sisters?It can’t be fishing rights, the Manderlys are doing fine in that regard while the Sistermen are better known for their shipwrecking. I also don’t remember reading about any attempted invasions by the Vale using the Sisters as a springboard. So why the 1000 year war?

It’s much more about naval control over the Bite than fishing rihgts, and the Worthless War came about in the wake of a series of naval conflicts between the North and the inhabitants of the Three Sisters:

“Even before the coming of the Andals, the Wolf’s Den had been raised by King Jon Stark, built to defend the mouth of the White Knife against raiders and slavers from across the narrow sea…During the wars between Winterfell and the Andal Kings of Mountain and Vale, the Old Falcon, Osgood Arryn, laid siege to the Wolf’s Den. His son, King Oswin the Talon, captured it and put it to the torch. Later, it fell under attack from the pirate lords of the Three Sisters and slavers out of the Stepstones. It was not until some thousand years before the Conquest, when the fugitive Manderlys came to the North and swore their oaths at the Wolf’s Den, that the problem of the defense of the White Knife—the river that provides access into the very heart of the North—was resolved with the creation of White Harbor.

…The last isles to be wedded to the Vale were the Three Sisters. For thousands of years, these islands had boasted their own cruel kings, pirates and raiders whose longships sailed the Bite, the narrow sea, and even the Shivering Sea with impunity, plundering and reaving as they would and returning to the Sisters laden with gold and slaves. These depredations finally led the Kings of Winter to send their own war fleets to seek dominion over the Sisters—for whoever holds the Three Sisters holds the Bite.”

So it’s absolutely the case that the Three SIsters *used* to be pirates and slavers, and shifted to becoming wreckers when more powerful forces cracked down on them, although they clearly still do a bit of piracy, because Stannis hanged twelve Sistermen for piracy when he was Master of Ships.

As for attempted invasions, well, where do you think Osgood and Oswin Arryn based and resupplied their ships from when they attacked the Wolf’s Den?

Is Myranda Royce snobbish about the Sistermen in TWOW because they’re poor, or because they’re not integrated into the Vale political polity, or some other reason?

Good question!

I think there are a couple different factors going on:

The Sistermen are not part of the mainland but rather out on the extreme periphery of the Vale:

Because the Sisters became part of the Vale after the Battle of Seven Stars, they don’t have the cachet of having fought for the Andals and the True Religion (in the same way that a certain kind of WASP looks down on people who didn’t come over on the Mayflower or whose ancestors didn’t fight in the American Revolution) and are instead provincials.
Because they live far off, they’re not a regular part of court life, and although definitely the highborn families would be treated as highborn, they certainly wouldn’t have the cachet of the inhabitants of the Vale proper.
Because they don’t live on the mainland, they don’t share mainlander concerns about mountain clans raiding or growing seasons or mountain passes being closed off by snow, etc.

The Sistermen are moreover a historically rebellious and quasi-criminal part of the periphery.

Before the Andals came, the Sistermen were pirates and slavers, neither activity likely to make them popular with the people they steal from or the people they steal. And to the present day, a big part of their economy is smuggling, and wrecking – the mainlanders despise honest merchants, so these guys are going to be seen as the lowest of the low.
As the mainlanders see it, the Sistermen started the Worthless War by being greedy pirates who poked the wolf one too many times, and not a small part of the mainland suffered as a consequence.
Then after the Arryns rescued the Sistermen, they rebelled against them during Aegon’s Conquest, and then against them and the Targaryens during the Blackfyre Rebellions.

And finally, there’s the whole inbreeding/webbed fingers and toes thing, which probably means they’ve interbred with the Squishers…

Is the Royces’ status as a principal house of the Vale at odds with their seeming pride in their First Men heritage? In a country noted for the value placed on noble Andal blood, whose First Men were mostly dispossessed and driven to become barbarians in the mountains, you’d expect any remaining First Men noble houses to try to assimilate into the Andal order – but things like ancient runic bronze armor seem to suggest the maintenance of a strong, persistent and distinctive First Men identity.

Actually, no, I don’t think it does.

I see the Royces as something of the Vale’s Loyal Opposition. They’re one of the most powerful Houses in the Vale other than the Arryns, they’re older than the Arryns (and Valemen are super-snobby so this counts), and no one challenges the Arryns except for them. Hence why a Royce cadet branch ended up at the Gates of the Moon, why we see Royces as Lords Regent of the Vale, why Royces marry into House Arryn (and I would imagine vice versa), why it was the Royce of Runestone who besieged Jonos Arryn when he turned traitor, and why Bronze Yohn was one of the first to turn against Littlefinger.

I see them as somewhat assimilating and somewhat accomodating. On the one hand, the Royces have converted to the Seven and are definitely bang on side with the whole idea of knighthood and tourneys and the like. So they’ve done the big ticket things that matter. On the other hand, they’re very proud of their First Men heritage – the runic bronze armor, the marriages into House Stark, etc. – but I think in a way that works for the Vale. Being arch-traditionalists, the Vale are very into heritage, and the Royces being Seven-worshipping knights means that the First Men stuff is made “safe.”

Maester Steven, may I please ask why you describe the Mountain Clans at the outskirts of the Vale as “oppressed” as well as dispossessed? (the latter is objectively true, but given that the Clans outright reject any connection with the Vale that doesn’t involve preying on the local peasants one would argue “Marginalised” or “Exiled” is a more accurate term).

Because it’s standard policy for the knights of the Vale to mount punitive expeditions against the Mountain Clans of the Vale:

“Before that, the chronicles tell of countless battles with the savage mountain clans.” (WOIAF)

“The mountain clans were lawless brigands, descending from the heights to rob and kill and melting away like snow whenever the knights rode out from the Vale in search of them.” (AGOT)

“The clans have grown bolder since Lord Jon died,“ Ser Donnel said. He was a stocky youth of twenty years, earnest and homely, with a wide nose and a shock of thick brown hair. “If it were up to me, I would take a hundred men into the mountains, root them out of their fastnesses, and teach them some sharp lessons, but your sister has forbidden it.” (AGOT)

If we do a bit of de-bowlderization, I think it’s fair to conclude that “root them out of their fastnesses” refers to cavalry raids against camps and villages of civilians, and “sharp lessons” refers to massacres. Hence why the mountain clans have to be described as “savage,” “wildlings,” and the like; de-humanization is required in order to rationalize the gap between the ideals of knightly conduct that the knights of the Vale espouse and how they behave at home.

How fertile is the Vale?

You may have already covered this, but I wonder exactly how fertile is the Vale? According to the wiki: “The Vale proper is a tranquil land of wide rivers, and hundreds of lakes. Wheat, corn, barley, pumpkins, and fruit grow in its fertile soil.”

By all accounts the Vale seems to be more bountiful than the North and probably the Westerlands. Yet for some reason the Vale can only muster about 30,000 or more soldiers at its peak, which is on par with the North.

I find this a bit peculiar because though the Vale proper cannot compete with the Reach and Riverlands in producing as much food (according to your estimates the Riverlands could boast an army of 80,000 at its full potential), surely it should still be able to sustain a larger population than the North and therefore be able to raise more troops.

I’ve discussed this here. The issue you’re missing here is one of size; the Vale proper is quite fertile (“a tranquil land of rich black soil…even in Highgarden the pumpkins were no larger nor the fruit any sweeter than here”), but it’s not very large compared to other regions of Westeros (”Though the Vale itself is famously fertile, it is small compared to the domains of other kings (and even some great lords), and the Mountains of the Moon are bleak, stony, and inhospitable.”)

Take a look at the map:

That blue circle is the Vale proper; the rest of the Vale is all mountains. That little triangle is just about the size of the Trident, far smaller than the Riverlands or the Westerlands as a whole, let alone the huge expanses of the North or the Reach. So while the Vale is quite fertile on a per-acre basis, and far more so than the North, the North has so much more land than the Vale that it evens out.

(Incidentally, I said the Riverlands at full strength ought to be able to raise 40,000, not 80,000 men).

Lysa, as a foreigner and a female leader, and an unstable one at that, seems to engender an unusual amount of loyalty from the lords of the Vale in refusing to enter the war of the five kings. Is this a result of Littlefinger bribing lords or something?

We get a pretty straight-forward answer to this:

“Others believe that Lysa must marry again, and soon. Already the suitors gather like crows on a battlefield. The Eyrie is full of them.“

“I might have expected that,” Catelyn said. Small wonder there; Lysa was still young, and the kingdom of Mountain and Vale made a handsome wedding gift. “Will Lysa take another husband?”

“She says yes, provided she finds a man who suits her,” Brynden Tully said, “but she has already rejected Lord Nestor and a dozen other suitable men. She swears that this time she will choose her lord husband…it seems to me Lysa is only playing at courtship. She enjoys the sport, but I believe your sister intends to rule herself until her boy is old enough to be Lord of the Eyrie in truth as well as name.”

The Lords of the Vale follow Lysa Arryn up until she marries Littlefinger, because many of them were hoping to win the competition for Lysa’s hand, and through her to win the Regency of the Vale.

How do you think the Manderlys managed to flee to the site of White Harbor, which is on the eastern seaboard of the North? As far as I know, the Mander doesn’t link with any river draining into the Narrow Sea, & the Reach borders the Sunset Sea not the Narrow Sea, so wouldn’t it have made more sense for them to flee to somewhere on the Stony Shore or Sea Dragon Point? Plus, why did they flee all the way to the cold North instead of say, the more verdant Riverlands or Vale?

Well, they probably sailed east around Dorne and then up the Narrow Sea.

As to why not the Riverlands or the Vale…the Riverlands is famously fractious and prone to private wars over land, why would any houses stand by and let some king take their land and give it away to a Reacherman when the Reach had been invading the Riverlands since the fall of House Justman? And what Riverlander king would have the authority to make that stick?

Likewise, the Vale is pretty small and the lands had been divided up early on, which is one of the major reasons why the Andals had invaded the Riverlands with the encouragement of the Arryns – not enough space, need to get people to emigrate.

Whereas the Starks had vacant land, because the Wolf’s Den was not being properly held and had been repeatedly attacked by the Arryns and the Sistermen and slavers from the Stepstones. The Manderlys offered an opportunity to solve a major security problem without the Starks themselves having to pay for it.