All three of those work fine. The king appoints whomsoever he wishes, baby. Additionally, if the capital city (and surrounding towns, I suppose) need to be reasonably equal for your background to work, then they will be. 

In general, the kingdom's policies will depend on which characters join the game. If you play high priestess of god X and bffs with the king, then god X is what the king will worship as well. Unless it's something unkingly, like a god of parliamentary democracies. That wouldn't work.