For final state acceptance one needs two states in general. The reason is simply that one needs to distinguish accepted and not-accepted strings. That is why single state PDAf are not a very useful concept. But if you have one, the prefix of every accepted string will itself be also accepted. These languages are prefix-closed.
If a deterministic automaton with empty stack acceptance indeed accepts a string, there is no longer string that can be accepted. The reason is that with empty stack in the classical model of Hopcroft&Ullman the automaton blocks, no steps are possible. Here determinism is important because nondeterministic automata just may have alternative computations on the same string. These languages are prefix-free.