How to show that pda accepts empty language?
I am not aware of any straightforward algorithms. One roundabout way I can think of is:

Convert the PDA to a CFG (there is a standard construction for this covered in introductory automata classes, the resulting CFG has $O(|Q|^2)$ variables where $Q$ is the size of the PDA)
Check if the CFG's language is empty.

We can check if the language of a CFG $G$ is empty using an iterative algorithm that marks a variable if it can derive a string:

Mark all terminals in $G$ (or delete them from the rules)
While no new variables are marked:

For each rule $A \to \alpha_1 \ldots \alpha_k$, if $\alpha_1, \ldots, \alpha_k$ are all marked then mark $A$.


If the start variable is marked then $L(G)$ is nonempty, otherwise $L(G)$ is empty.

The algorithm above can be optimized with a worklist that skips re-visiting a same variable's rules after it's marked, and doing some processing to allow checking rules affected by the previous pass