yes i did ballet it takes alot of hard work i was so sore after every ballet practice but thats probably because i took a hip pop dance class then a hour break then ballet but its hard and its definitely consider a sport kinda just like gymnastics but kinda not in some ways but i would still consider it a sport
yes, Martial Arts training is an intensely difficult sport, and as every bit of ballet I have ever seen, requires just as much grace and fluid motion as Martial Arts, if not more so. Then it is most definitely a sport, and a challenging one at that.
Unless there is a competition component to it, then no it's not. It's a form of art and expression. This isn't saying that it's not rigorous or demanding, or that it doesn't take a lot of hard work and practice, but to be defined as a sport - there has to be an actual competition involved. (and no auditions don't count XD)
I have done ballet along with tap, jazz, and lyrical since I was four. It is definitely a sport. You feel some pain, you sweat, and you compete. It is also a form of art, but I definitely count it as a sport. Ballet has a ton of posture and technique involved; I wish you the best of luck at it and remember that if you can succeed at it, you are extremely talented!