I use them all- Tamiya, Gunze, and MM Acryl. For thinning, I use a mixture of 25ish% isopropyl alcohol, and 75ish% distilled water (or tap water, when I'm lazy). Mix this ratio by your preference. More alcohol, and the paint will dry faster, more water and it will dry slower. Too fast (most or all alcohol), and the finish may be grainy, as the paint will partially dry in the spray stream. Too slow, and its of course more likely to puddle- but, I've used anywhere from 1:3 to 1:6 alcohol:water and had good results- so this ratio isn't too critical. I keep this ina small plastic dropper bottle- I pour a bit of paint into the airbrush paint cup, and put a few drops of thinning solution in, test spray, and adjust, if necessary.
For cleanup, I use windex glass cleaner- don't use the cheap generic stuff, it doesn't have enough ammonia in it to be effective. Just put it into the paint cup, and spray and backpressure until it runs clear. Windex works very well as a stripper for Tamiya and Gunze acrylics (so be careful using it around painted objects!), and it's terrific for cleaning the airbrush. Windex won't dissolve
dried Acryl, but it works great for cleanup after spraying. If you leave it to dry, Testor's MM Acryl cleaner will certainly do the trick. It comes in another small plastic dropper bottle, just like I use for thinning solution and windex.
Before painting a car body, or anything else that I'm super paranoid about paint contamination on, I take the airbrush apart and clean the tip, needle, paint cup and siphon with a bit of laquer cleaner, with qtips and pipe cleaner. Laquer thinner will clean just about anything that may be dried in your airbrush.
Your right that gloss acrylics do take a while to dry until they're ready to polish, but I've never had a paint job with acrylic that wan't ready to polish in about a week. I've had enamel finishes take up to six weeks to cure. Frequently when I polish, I find that I can spray a couple of thin coats of Tamiya clear (X22 I think), polish it after two days, spray some more, wait two days and polish some more(repeat as necessary). I find that it's very forgiving, and a little bit of patience will yield a blinding P-man shine.
Give it a try!