I searched all over the web, and I could only find a few others with this problem, but no solution.

Well, the problem is as the title suggests:
In Skirmish games, the (Re)move and the Save scene buttons are grayed and crossed out.
Without being those two options, building bases and whatnot are a real pain-in-the-ass in Skirmish mode.

So I'm wondering; why those are disabled? The options seem to be there, and bases are being saved in the campaign...

It's on purpose, you can only save scenes in the scene editor, and you can't remove stuff either. I don't remember why you can't remove things, but there was a reason. Yes, building bases is repetitive and boring, so I tend to stick to the prebuildt bases (entire bases as one module) from Unitec and that huge module pack. You still have to place actors and weapons, but at least you don't spend loads of time building a new base every time you play skirmish.