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.

I’m always confused on whether we know when Littlefinger learns he slept with Lysa instead of Cat. Do we have any indication he knows this before the moon door scene?

That’s an excellent question, because I realized that I was also unsure about this question. My asearchoficeandfire-fu was failing me, but @goodqueenaly found this rather interesting passage from Tyrion III of ASOS:

“My lords, with your leave, I propose to travel to the Vale and there woo and win Lady Lysa Arryn. Once I am her consort, I shall deliver you the Vale of Arryn without a drop of blood being spilled.”

Lord Rowan looked doubtful. “Would Lady Lysa have you?”

“She’s had me a few times before, Lord Mathis, and voiced no complaints.”

It’s a rather surprising moment, because I had been looking in AGOT for where Tyrion discusses Littlefinger’s boasting at court, and had had a false memory of him boasting about both Tully girls when in fact the text only mentions Catelyn.

So this raises some interesting possibilities: either Littlefinger had realized what happened at Riverrun, but was unwilling to admit the full truth either to himself or anyone else, or he and Lysa had actually had an affair either in the Vale or King’s Landing (which is something that’s been debated but never proven), or Littlefinger was inadvertently telling the truth while lying.

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.