So, it just occured to me that Gregor Clegane doesn’t appear to have a wife or family. Although I’m really glad that for that(as a human being with empathy and feelings), isn’t it a huge dereliction of his duty as the head of a house? Shouldn’t that be addressed in the narrative?

Actually this has been addressed in the text:

Ned Stark could not recall ever speaking to the man, though Gregor had ridden with them during Balon Greyjoy’s rebellion, one knight among thousands. He watched him with disquiet. Ned seldom put much stock in gossip, but the things said of Ser Gregor were more than ominous. He was soon to be married for the third time, and one heard dark whisperings about the deaths of his first two wives. It was said that his keep was a grim place where servants disappeared unaccountably and even the dogs were afraid to enter the hall. And there had been a sister who had died young under queer circumstances, and the fire that had disfigured his brother, and the hunting accident that had killed their father. Gregor had inherited the keep, the gold, and the family estates. His younger brother Sandor had left the same day to take service with the Lannisters as a sworn sword, and it was said that he had never returned, not even to visit. (Eddard VII AGOT)

Gregor murdered most of his family and all of his wives.  

2 ==> If breaking the King’s Peace is treasonous, did Catelyn commit treason by either abducting Tyrion or after abducting/arresting Tyrion, by not taking him to Kings Landing to stand trial? 3 ==> Even if it wasn’t treasonous, what would Catelyn’s punishment have been for not taking Tyrion to KL, especially had he been executed thanks to a quick step out the Moondoor? Who would be held accountable? Stark, Tully, Arryn, Catelyn, Lysa… (assume no boars or incest to muddy the KL environment.) TY

2. Arresting Tyrion isn’t a breach of the King’s Peace. Catelyn was very careful to follow the legal forms and would have had dozens of witnesses that she had arrested him for a crime committed in Winterfell and was bringing him there to await trial, as is Eddard Stark’s right as a Lord possessing the right of pits and gallows. 

3. She probably wouldn’t face a penalty, given the ambiguity of royal justice and the right of pit and gallows. Especially without evidence of much in the way of royal courts, there’s not much of a sense that there is a right to be tried in a royal court as opposed to by a lord. 

1 ==> Is breaking the King’s Peace an act of treason agianst the Crown?

It depends on how the authorities want to treat the case, and what kind of act we’re talking about. Within the Westerosi tradition, we know that Eddard Stark as Hand of the King declared Gregor Clegane’s attack on the Riverlands worthy of attainder, which is associated with serious felonies and treason, and that historically kings like Aegon V have dealt with breaches of the peace by leading royal armies in the field to arrest and quite likely execute the violators. 

On the other hand, in the English legal tradition, making a “breach of the peace,” wasn’t technically a criminal or civil offense, because it’s considered a violation of the royal prerogative to maintain the peace. So what happens when you commit a breach of the peace is that you get arrested, put in front of a magistrate, and “bound over” (essentially put on probation/bail) where you are required to refrain from certain activities (usually but not always the activity that involved the breach) for a given period of time. However, if you violate the terms of your binding, you’re in contempt of court, and then criminal penalties involve. 

However, a lot depends on what kind of act we’re talking about. A drunken fistfight is technically a breach of the peace, but so is banditry. And given the context of Ser Gregor Clegane’s case, he was being accused of banditry, murder, wanton destruction of property, and in short being an “outlaw.” And in early common law, if you defied the laws of the realm you could be declared an outlaw or a legal non-person – which meant that murdering you was legal, and helping you was a crime. It’s arguably worse than being a traitor, as legal penalties go.

Likewise, if the powers that be decided that your breach of the peace was made as a deliberate insult towards the king – that you were implying through your action that the king was too weak or feeble to defend the peace, for example – you might be guilty of Lèse-majesté, and that is a form of treason. 

In a hypothetical scenario, if sometime, during the reign of the Iron Throne over the 7 Kingdoms, a member of a Paramount House A publicly abducted a member of Paramount House B, would House B be breaking the Kings Peace by taking military action to retrieve said person from A’s captivity? Is the answer still the same if you take the example of what happened in the Sworn Sword between the Webbers and Osgreys? Enjoy.

Wow, what an unexpected and unusual hypothetical scenario…

The answer is yes. Taking military action against another house, instead of going to the law, is the very definition of breaking the King’s Peace. 

Tywin left behind only 3 children. That seems like a very small family. How large was the average Medieval noble family? And why did he never remarry? Surely he’s not a sentimentalist?

Well, I’d direct you over to @joannalannister who would explain that Tywin was very much a sentimentalist when it came to his wife. 

On a more practical level, Tywin had dozens of nephews and nieces and cousins, so House Lannister wasn’t hurting for heirs. 

Thanks for the Golden Tooth reply. As a follow up: Stannis said that Aegon saw Westeros as “One realm, for one king to rule alone”. My Q is not about the Pros vs Cons of Aegon’s Draconacy, but about Aegon’s specific Draconic doctrine. You suggested that he could have claimed the GT as a crown possession. Expanding on this wouldn’t an ideal doctrine then be to further control inter-paramount politics by also claiming the Bloody Gate, Moat Cailin and **Skyreach. Your thoughts are eagerly awaited.

If Aegon didn’t have the dragons, that would be a pretty good policy if your objective is to unify previously warring kingdoms and prevent interregional warfare. 

However, I firmly believe that Aegon felt the dragons obviated all of that. Harrenhal was all the proof he needed that fortresses meant nothing, when he could rain fiery death from the sky at will and, more importantly, when everyone in Westeros knew what he could do. 
 

Is it likely that there is a significant portion of the peoples of the North that migrate to the southern kingdoms (especially after Westeros’ unification) with the onset of winter or does the shorter agricultural windows during the climactic cold periods Planetos calls winter especially necessitate all hands on deck for the communities to survive, adding to the benefits of the Winter Town System? Thank you.

Well, we do have an example of such a migration, in the case of Cregan Stark’s invasion of the south in 131 AC:

Nowhere in the Seven Kingdoms did the winter matter more than in the North—and the fear of such a winter had driven the Winter Wolves to gather beneath the banner of Lord Roderick Dustin and die fighting for queen Rhaenyra. But behind them came a greater army of childless and homeless men, unwed men, old men, and younger sons, under the banner of Lord Cregan Stark. They had come for a war, for adventure and plunder, and for a glorious death to spare their kin beyond the Neck one more mouth to feed.

The day after the executions, Lord Stark resigned as Hand. No man ever held the office so briefly, and few left it as gladly. He returned to the North, leaving many of his fierce Northmen behind in the south. Some wed widows in the riverlands, others sold their swords or swore them in service, and a few turned to banditry.

So it does happen, but it’s a rare phenomenon that takes both an extreme winter and a prolonged war. 

If you were Aegon the Conqueror, would you have included House Lefford of the Golden Tooth into the Riverlands under House Tully’s Lord Paramouncy or kept them as part of the Westerlands? (P.S. I’m asking Attewell, SLAL and the Good Queen, having huge respect for all y’all’s differing POVs, TY)

Well, it sort of depends on how, as King, you view the loyalty of the Westerlands and the Riverlands. After all, the whole point of Aegon’s Conquest was to assert the unity of Westeros and establish a scenario where wars between kingdoms don’t happen, and given that the only purpose of the Golden Tooth is to ensure that the Westerlands can’t be invaded by the Riverlands but can invade with impunity, the King’s rule should make the Golden Tooth irrelevant. 

So the only scenario where I would see Aegon giving the Golden Tooth to the Riverlands is a scenario in which the new king sees the Riverlands as more loyal than the Westerlands, and needs to forestall any future rebellion which might involve an invasion of the Riverlands. 

Indeed, if it wasn’t for Aegon’s dracocracy leading him to make minimal changes to pre-existing conditions, it wouldn’t be a terrible idea to annex the fortress to the crown to prevent either kingdom from invading the other without openly rebelling.