Multiple homeworlds is akin to multiple players - The second homeworld doubles the AI's strength, the third makes it 3x of normal, etc. With that said, there is a large group of people that wont play without 3+ homeworlds.
I consider anything smaller than an 80 planet map 'too small', and anything greater than 1 homeworld starts blasphemous. I have beaten difficulty 8 AIs on an 80 planet map with a single homeworld. I would probably not say 'easy', but certainly not impossible.
Champions are kinda weak until you spend some time 'levelling' them. Either through nebulas, or alt- progression (I've been meaning to do a game of that sometime, so I dont know specifically). Generally if you have access to a nebula, you should try to visit it. Until you do a few of them, the champion wont be anything particularly noteworthy.
List of all the ships available? The wiki should have most, if not all of them, mostly up to date:
Here. That might be your best source. Remember you can pause the game at any time to look at what is in play and think of how to deal with it.
Examples of difficult stuff? I tend to really like the no-wave-warnings, cross-planet schizophrenic waves, on diff 8 or so. Dont know when waves are coming, where waves are coming from, dont know where they are going, and they often decide not to bother attacking and just become threat. Really emphasizes scouting to keep up with the AI. Also, hybrids tend to really muck things up. I think I am the only person to go quite as heavy on the wave modifiers, but others do some crazier stuff with flat difficulty. Usually this doesnt kill me, unless the waves happen to coincide with other major events (which I'm pretty sure keith recently put in code to force more often)