Tyrion mentions that the Westerlings had sold off a large portion of their lands. How exactly would such a transaction take place in a feudal economy? Would there be restrictions on who they could sell to and for how much?

Discussed somewhat here

The Westerlings selling their land is a highly unusual event in Westeros – the only other times we hear about selling land is in the context of the Tarbecks forcing people to sell their land through threat of armed force, so voluntary (to the extent that the necessities of poverty qualify as voluntary) land sales are a sign that the feudal order is in crisis. 

It suggests that the Westerlings were falling into genteel poverty, such that their rental income had fallen massively behind their ability to service their debt, and that they were having to surrender the collateral they had put up to secure the loan. 

Legally, this could be quite tricky. In Medieval England, for example, the feudal principle of “Nulle terre sans seigneur” (no land without a lord) meant that selling land outright, known as “alienation of lands by will,” was actually legally impossible until the late 12th century. (The Magna Carta, for example, says that “No free man shall henceforth give or sell so much of his land as that out of the residue he may not sufficiently do to the lord of the fee the service which pertains to that fee.”) Selling land was legalized by the Statute of Quia Emptores in 1290, although the buyer was “required to assume all tax and feudal obligations of the original tenant,” so the land remained under the same lord as before. It wasn’t until the Tenures Abolition Act of 1660 that those feudal obligations were eliminated. 

Would you guess that Janos Slynt was born with his surname (which would be unusual as his father was a lowborn butcher), or somehow acquired it or invented it later in life?

If it’s anything like medieval England, the surname is due to the fact that there’s a half-million people living in King’s Landing and you can’t do what you do in a small local village, which is to distinguish between two people with the same first name on the basis of you know who’s related to who (so it becomes “Tom, Bill’s cousin” or “Tom, Sarah’s nephew”). 

However, it could be that Slynt picked up the last name when he became Captain of the Iron Gate or Commander of the City Watch, both of which are important enough posts that he might be considered worthy of a surname. 

How much power does a crown prince actually have? Like with Joffrey for example, if Robert had lived and the whole incest thing was put off a while and Joffrey was a man grown, but not named protector of the realm or hand of the king whatever title a crown prince can hold, and did a tour of one of the kingdoms, and a lesser lord or a knight did him wrong, whether it be verbally or physically attacking him does he have it in his power to make a decision on what’s to happen to said person

How much power did Aerion have over Dunk?