Drunken vices are just random, AKAIK Inns and Tavern have no effect.

As for health, 3 health points give +1 hit point. Melee and ranged effect health the same way, however for the larger artilery pieces you can usually kill the strongest king in one shot if you get a direct hit (good luck!). Lives refresh after every battle, although if they lose health without dying they have a chance to develop some vice and virtues (such as hideous scars). Kings and princes do indeed get extra health, kings more so than princes. I am unsure whether they still keep this bonus when they become normal generals, but I think so. Regular generals get a small health increase, not sure of the exact amount though (5 I think? I'm going to say 5). You can really rack up health if you send your king into battle, with the bonus health for being a general, the king bonus, plus and vice and virtues (such as mighty warrior which is common amongst heirs) you can have a truely Jedi general!