It looks like fun, and requires a lot of skill. It has more in common with gymnastics than combat.
Yes, the main difference is intent. Martial arts should (with some notable exceptions) focus on efficient movements designed to do specific things (injure, break balance, block, evade, etc.). Gymnastics should focus on showing what the human body can do with balance, high coordination, and strength. Dance is typically focused on grace, though much urban dance incorporates a lot of the principles of floor routines from gymnastics.
That depends upon the criteria you use to define "valid". Some would say MA are only "valid" if they are combat-tested (which rules out nearly all). Some will say only if they are fight-tested (which typically rules out all but sports). Others will say they must be focused on defensive combat (which rules out most sport).
In my opinion, they are all three quite different. Very different biomechanics, to accomplish different things, even if they appear similar on a superficial level. In my experience, dancers can have some real difficulting training into martial arts. They can't Make the switch in the biomechanics. They also get frustrated if they are an accomplished dancer, they can "do the moves" of the martial art, but it's still all wrong. Some of them don't like being told that.
Don Angiers (quoted previously) told me he preferred his students to have dance experience, because they'd have better balance, grace, and ability to keep their weight centered. At the same time, I can see where an experienced classical dancer might have issues, because their mechanics are mostly around balance and appearance - if a dance move looks right, it probably is. With MA, looking right and working are not synonymous.
Yeah, modern dancers have the problems too. That dance background gives some real advantages, but also some very real disadvantages. It can be a real mixed bag. People assume it would be a benefit. Sometimes it is, but sometimes it is a detriment.
