Any thoughts on the fact that the prologue & epilogue POVs up until ADWD are largely defined by their powerlessness, while ADWD features a powerful skinchanger who acts like a grotesque parody of the worst of the Westerosi aristocracy and the King Regent, nominally the most powerful man in the Seven Kingdoms?

I wouldn’t call Varamyr in the Prologue of ADWD as a powerful figure as a figure who has been brought low, desperately scrabbling to avoid true death and failing. 

You say that the Starks derived power from the fact that they enforced justice in an almost legalist manner (as much as a feudal society can be legalist). But you also say that there’s not much difference between Northern and Southron politics. Yet Ned’s application of the Stark style of justice and ruling doesn’t jive with the Southron court. What gives? How is Northern politics simultaneously accepting of Stark judicial authority yet also close to Southron politics?

Not that different on the specific modality of Northern lords being just as focused on scheming for power as their Southron counterparts. After all, this question emerged from the question of why Ned fails as Hand of the King and whether that emerged from a qualitatively different form of Northern politics.

Northern judicial authority doesn’t really have relevance for Ned’s tenture as Hand. 

Would a lord of the Vale have any chance of gaining the fealty of the mountain clans in the same way the lords of Winterfell did for their own mountain clansmen? Or is the cut too deep?

No. The hill clans of the North were integrated within the political community from practically the earliest days after the Long Night. The mountain clans of the Vale were dispossessed and oppressed for six thousand years by the Andal lords of the valley. 

Why do you think Jon Arryn took Hugh as his squire? Hugh didn’t seem to be from a noble family.

He’s not from a high-ranking family with a recognizable name, but he could easily be the son of a household knight or other retainer from the Eyrie. Certainly Ser Hugh was poor and without prospects other than Jon Arryn and becoming a tourney knight, but that’s not necessarily a good metric of whether or not someone is highborn or lowborn. 

Do any houses in the westerlands(current houses, excluding cadet Lannister houses) have a substantial amount of gold from mining like the Lannisters- maybe the paynes?

I would say that the norm is that Houses with their holdfasts in the hills (as opposed to say Crakehall or Cornfield) have a good deal of income from mining precious metals (mostly gold, but not entirely, hence House Serrett’s seat being named Silverhill), unless otherwise indicated. (”The Westerling mines had failed years ago.”)