2.) What do the characters do?
Make arrangements with the aforementioned Gods, and attempt to become more than pawns on the chessboard of faith, politics, and where those ideas collide. Also, they transport (read "smuggle") Gods and holy objects under the evil Emperor's watch, possibly even to the extent of gaining power to overthrow the Emperor himself.
The "standard" (by which I mean, "described and suggested") character types for Illegal Gods are: Pirates, Ambassadors, various members of Military Forces, Spies, Asteroid Miners, Priests, God Merchants, Soul Jockeys, Demigods, Scientists, Bureaucrats, and Droids. Most of these are fairly self-explanatory, but there are some professions (God Merchant, Soul Jockey) that are unique to the setting.
This also implies that PCs will have things the Gods will want, so they can make those arrangements. I also am looking to have more faith and political conflicts in scenarios rather than, say, treasure hunts. (Although because I happen to like dungeon crawls and exploratory scenarios, I don't want to close those off entirely.) I don't know how much this means I need to fabricate the political spectrum, although I do have quite a number of seeds to suggest.