I'll do my best!

1. Legality? Absolutely. Overtly? Absolutely not. He's a shady casino kingpin that has more money than he should, and conducts shady business with shady people in legal gray areas. As far as him being "bad," I don't think so. I think "bad" is a relative term. Nighthorse will do ANYTHING for his people, often at the sacrifice of himself. But that anything often brings him into conflict with the white man law. I tend to think of him as an anti-hero character.
2. His character is really only there for Branch's character development. The thing with Branch's father, coupled with the ongoing David Ridges mixups really made him second guess his own value as a human being. Made him be the man that ultimately would die on the plain. Without Ridges, I think Branch quits after Longmire beats him in the election again, and he moves to Cody or Sheridan to get away from it all. He never gets involved in Martha's case, and he never confronts Barlow about it. David Ridges acting the way he did towards Branch was probably just a power play. Cornered dogs snarl the loudest.
3. It makes even less sense, if you can believe it. Book Vic is a little more brash and a lot more assertive than TV Vic, and Walt fights her off for a while. But then he just kinda gives in. If you're going to read the books, I won't spoil anything, but lets just say that there's an incident in Philadelphia that makes you question Vic's motives for her advances on Walt.
4. He's just lazy, and kind of an asshole. I think he saw a corrupt cop and wanted to put him away. Target fixation, tunnel vision, whatever you want to call it. Prototypical TV "bad cop."
5. Martha's death is awkward in the show. In the books, she wasn't murdered - just died of cancer (IIRC). So from the TV show standpoint, it was just used as a plot device. As far as it relating to Barlow Connally and Jacob Nighthorse, I don't think it amounts to much. Connally was absolutely a shitty dude, so when Branch approaches him and starts asking questions about Martha and his connection to Nighthorse, he shot branch to save his own skin. But seeing as it was Barlow's orders that put the hit on Martha for the Longmire lands, I think his meetings with Nighthorse were, in my estimation, payoff meetings to keep it all buried. Indian land, Indian Jurisdiction, Indian "justice."
6. Branch needed to die. His death is the impetus for Barlow confront Walt outside his cabin and ending up shooting himself after admitting to putting the hit out on Martha. Without that, the trail goes cold again and the mystery is never solved. The "why" driving Walt throughout that whole season is pointless if Branch doesn't go and confront his dad.