In regard to what Favor does: I did a bunch of test runs and data gathering last spring, and with a variety of builds the main thing I discovered was this:

Favor of 90: Ships got two "rolls" (could bring back two items from each sector visited)
Favor below 90: Ships got one roll per sector

My conclusion was that a high favor gave more "bites of the apple", so to speak. I can't tell if it affects the odds that each roll will get you something very rare, but it does appear to determine how many rolls you get.