How many Catholic rites are there, and can a Catholic attend them, receive communion, or become a priest in them?

That is a good question. I think there are about a dozen major rites, not that many, though there could be more; I may be off. Yes, as long as the rite is recognized by the Church in Rome, you can receive communion and all the sacraments there, whether it is Coptic, Ethiopian, Assyrian, or whatever, all kinds of them that cropped up in the Middle East during the first centuries of the Church. So yes to your question.