Viewing the Southern Cross from Key West

Is it possible?

Is it a year round thing or possible only during some months?

Yes, at the right time of year, and at the right time of night:

According to that, it may be seen at the right time any place south of 26 degrees north, which includes southern Texas and Florida - Key West is about 24.5 N. You won’t see the whole thing for very much of the year, and it’s going to be down by the horizon.

Does that mean I could go to Key West in, say December, provided that I was willing to get up at some inconvenient time to do so?

Actually, about now is the good time: