Yes it is, and it's a good question! In fact, 'my' is the grammatically correct form. 'Watching' is a gerund, and gerunds need to be proceeded by a possessive adjective, just as nouns are. 'Do you mind my watching...?' is 100% correct. The problem, though, is that it doesn't sound correct to many people living in English-speaking countries who haven't studied grammar. This includes both native speakers and second-language speakers. Colloquially, the majority of the English-speaking population would say 'Do you mind me watching..', even though this is - strictly speaking - grammatically incorrect. The 'me' form is now so common that it has become acceptable outside of formal contexts. The only other comment I'd make is that, although they're correct, the three 'ing' forms make the question sound a little awkward. If you wanted to say this in the real world (as opposed to in a grammar exercise), you could change it to 'Do you mind my standing here watching you dance?'. There's a subtle difference between 'dance' and 'dancing', but it amounts to the same thing in this question. Or you could omit 'standing' 'Do you mind me watching you dancing?' I hope that helps.

I think it is me not my but usually someone would just say "do you mind me watching?" or "do you mind me looking on while you (whatever they are doing?)"