Do they get any say in music or dance?
Not anymore, and certainly not as much as they did in the early series.
An friend of a friend was a contestant fairly recently. At one point they refused what they believed would be a dreadful theme/costume/song, but it took a hell of a battle and their pro had to fight hard against it. The crappy theme was apparently later done by someone else, who presumably was more amenable and/or had a less determined (or less interested) pro.
Sadly I don’t know any more about it as he wouldn’t give much detail away. I’ve tried to work out what it was on the basis of what awful themes came up later in that series but there are several contenders. 
But yeah, the impression he (discreetly) gave was that you don’t get much say in stuff. Though I’d guess if your casting is viewed as something of a coup you’d be allowed more autonomy.