Repair path - leaking bathroom drain and drywall damage
The drywall doesn't need to be replaced unless 1) it gets moldy and gross, or 2) its structural integrity is compromised. Get some fans on it as soon as possible, from above and below. If it dries out and doesn't feel mushy you can simply prime and paint. 

It's possible that the leak is at the tub connection, but it's also likely that it's in the trap assembly if it was put together with compression fittings (as opposed to being cement-welded). Do you have access to the drain area? If not, consider installing an access panel below or at the end of the tub. 

A plumber won't be able to tell you anything that you can't determine on your own. They'll need to see what's in there, too. Once you know, repairs can begin. It's probably not too difficult considering you're working with PVC and modern fixtures.