What should I look for in an indoor cycle trainer?
I can't recommend any specifics offhand, other than I've used two different types of Tacx trainers and have been very happy with them. Both were in the $150-$200 range new.

A couple of comments though:

a &amp; b: Virtually any trainer on the market is going to be pretty stable unless you are out of the saddle and sprinting. As long as it's not a really low end one if you are wobbling you might want to spend some time working on spinning smoothly. And most everything on the market will fold up a bit, though not tiny. Think something in the loose ballpark of the size of half a wheel.

c. If you're looking lower on the budget range, don't worry about not being able to switch the resistance mid-workout on the trainer. You can get a wide range of workout just by shifting gears on your bike. 

d. (you didn't mention this one) Noise level. Some of the wind and magnetic trainers are loud. If you can try it out, great. The tacx trainers I've used were relatively quiet. The Minoura one I've used was painful.