It's still part of Cuba, but it's on lease for undetermined term (to end it, both side would havd to agree or USA would have to unilaterally and completely withdraw). About jurisdiction:

Article 3 – Cuba retains ultimate sovereignty, but during the occupation, the U.S. exercises sole jurisdiction over the areas described in Article 1. Under conditions to be agreed on, the U.S. has the right to acquire, by purchase or eminent domain, any land included therein.

Of course as it is immediately obvious, it was an extortion and is really an occupation. Problem lies in that evicting the base would require military action and it would be a war.

Yeah probably. I guess it'd be more accurate to say there's a mcdonalds located on the island of Cuba though technically outside of its legal jurisdiction, but that doesn't sound as good.