How do I show that whether a PDA accepts some string w is decidable? or undecidable?
Given a PDA and a string $w$, there are several algorithms that decide whether the PDA accepts $w$. In fact, for a fixed PDA, such algorithms run in time $O(|w|^3)$ or less. The most well-known algorithm is the CYK algorithm. Although CYK expects the language in the form of a context-free grammar rather than in the form of a PDA, there is an algorithm that converts a PDA to a context-free grammar generating the same language.

I expect every decent textbook on automata theory to contain a proof that the acceptance problem for PDAs (given a PDA and a string, determine whether it accepts the string) is decidable, say along the lines above.