is determining an unknown CFL from intersection of two CFLs decidable?
It's well-known you can present computation history of a Turing machine as an intersection of two CFLs. Take a deterministic Turing machine $M$ and force it to reject everything except possibly the empty word. The set of computation histories is either empty (if $M$ rejects the empty word), or a singleton (if there's an accepting computation for the empty word), and if there was an algorithm for your problem, you could tell those cases apart.