What really has helped me was going for a lower-heeled shoe with a slightly thicker heel, and making sure I got something that was plenty wide across the ball of the foot. I was having foot and knee problems for a while until I figured out that I was wearing shoes that were too small! Sure, dance shoes are supposed to fit snugly, but there is "snug" and then there is "too tight." I'm wearing a size larger than I used to, now, and my feet and knees are so much happier. I put those non-slip heel grips in the backs of the shoes, and they're staying on just fine. I also don't wear shoes higher than 2" any more.
As far as support goes, I like having a decent heel cup and well-constructed heel, but I do not like having the soles or the parts that go around the balls of the feet be too stiff. In fact, I'm much more comfortable and dance better when that area is very flexible. As the shoes get worn and start to stretch, I'll put in a foam or gel half-sole.