I could be wrong, but I think if you wanted to ask where the puppy is dancing it would be "강아지는 어디에서 춤을 춰요?"
Getting into the grammar, dancing is an adjective here. "The dancing puppy" - dancing is modifying puppy. This is more clear in Korean (to me) because it uses the 는 adjective modifying particle after the verb "to dance" (춤을 추다) creating 춤을 추는.
The verb or the sentence is 있다 which is to be or to exist. We know this because it's conjugated as a verb and, in Korean particularly, because it's at the end of the clause.
The final thing we can look at is the location particle being used. In this sentence 에 is being used. This can mean to a place or it can reference something which exists in/at a location. It does not, however, mean the location at which an action is taking place. For that we use 에서. So the sentence is asking "where does something exist" not "where is an action taking place". That's how you know.
In the other sentence ("강아지는 어디에서 춤을 춰요?") we see a literal translation of "The-dog where-at dancing?" vs "Dancing dog where is?" The literal stuff is a little silly and you shouldn't think of Korean as English, but I thought it might help you see where the focus of the sentence is in Korean. Below are some links that might help clear up confusion about the grammar.
The adjective form Part 1: http://talktomeinkorean.com/lessons/level-3-lesson-13 Part 2: http://talktomeinkorean.com/lessons/level-3-lesson-14
Location particles: http://talktomeinkorean.com/lessons/l1l18/