Why do random Capital Ships (that I can control) just start showing up at my Homeworld later in the game?
Those are the hero units. They are spawned by a structure from your homeworld. I have to have them on autocast for the AI to use them, though if you're playing the Rebellion version the next patch that I'll be releasing soon will tweak them a little bit.
Also, is the mod balanced for the "competitive play" random maps, or should I just use the default type?
Okay I guess you are playing Rebellion. The competitive maps should work with the set resource extractors and militia, even on the new planets. They're are a couple of things the mod spawns called "Random Encounters" that may still be active on those maps that probably aren't wanted in competitive play, but that should be corrected in the next patch.
I've tried your mod, and I really like it so far, I just have a few questions.
Also for future reference, please post in the mod thread or on moddb for future questions, just to make sure I see them.