Does the BC exam only get you credit for Calculus I?

It doesn’t seem to cover all the Calculus II material.

It depends on the university but BC usually grants credit for calc 1 and 2. Some colleges, though, include what some might consider part of calc 3 in calc 2, in which case you may miss some material. But generally, BC covers all of calc 1 and 2.

As said above, it totally depends. Check the AP policy of the school(s) in question.