Can we push and pop both at a single transition in a PDA?
(Previously a comment)

Depends on how you define PDAs. The definition I consider canonical certainly does allow you to exactly this: it is assumed that each transition pops the top-most symbol, and pushes an arbitrary string. To represent not changing the stack, you'd push the same symbol you just popped; to pop one symbol, you'd push the empty string; to put $w$ on top of the stack, you'd push $wa$ (where $a$ is the top-most symbol); and to replace $a$ with $b$, you'd push $b$.