Suppose $L \subset A^*$ is context free and $A^*\setminus L$ is also context free. Does it mean, that $L$ is deterministic context free?
If it is not, I would like to see a counterexample (I failed to construct one myself).
Note that the converse is true. Moreover, a complement to a deterministic context free language is also deterministic context free as one can simply change labels on the corresponding deterministic pushdown automaton.
No, as shown in this answer and this one on cs.stackexchange. You cannot even conclude that your language is unambiguous, as shown in this answer.