Yes, that is all correct. I watched her dance. (no ~ing) I saw her dance (form beginning to end). I saw her dancing (in the middle of it). Almost the same, not quite. A large overlap, too.

When you say "I watched her dance" it has the feel that the event happened farther in the past. When you say "I watched her dancing" it is more like the imperfect past because it was happening for a while before you stopped watching or that the event is in the more recent past. I hope that makes sense.

as I know both are correct, dance is a noun, and dancing is a gerund that can replace the noun,