I was concerned with that as well, but found a post by Spinner at one point where he mentioned that he oils first and then applies a 50/50 mix of dye with the airbrush. The oil helps to pull the dye into the leather instead of leaving it on top. Doing that with a couple of coats, I've never had any issues.
For a thinner with the oil dyes (Fiebings Pro Oil) I use plain old Isopropyl alcohol with good results, but many people actually recommend using denatured alcohol since it's closer to the true base of the spirit dyes. I'm just cheap and I haven't had any issues with the Isopropyl at all.
As for the differences in the dyes (at least as far as my experience goes - I may be wrong on occasion):
Dye: Available in several different types - the most common being spirit dyes and water based. These dyes will actually penetrate the leather and color it, hence the previous discussion regarding the penetration of airbrushing dye. There are a few different types of spirit dyes, Feibings Pro Oil being my favorite.
Acrylic: These are more closely related to paint. The color doesn't really penetrate the leather and just sits on top of it, leaving it vulnerable to cracking and peeling. Some people have great success with them though. Personally, I'm afraid to use them for anything more than accents.
Antique: This is more like a shoe polish. It's meant to highlight the tooling and/or give an older appearance to the leather. It will color things, but I would think it would be hard to get an even uniform coverage. Usually you would use a clear finish over the entire piece and then apply antique so that it will settle into your tooling without coloring the rest of the leather. One more clear coat after that and you're good to go.