1. Look at the [product construction](https://courses.engr.illinois.edu/cs373/sp2010/lectures/lect_04.pdf) for DFAs, it will give you a hint of what the state space of a PDA accepting the intersection will look like.

2. You can covert $D$ to a PDA, but you have more information than that. It doesn't need to touch the stack at all, since it can be accepted by a DFA.

These should get you close to the answer.