Simple question: Pushdown automata transition
This depends on the formalism you are using.

As a "abstract data structure" you would have simple instructions like push(symbol), pop, top, isempty that you then can program as an add-on to a finite state automaton.

In standard formal language theory one uses a single instruction of the form you describe $(p,a,A,q,\gamma) \in \delta$ means: in state $p$ with $A$ on top of the stack we can pop $A$ and push the symbols $\gamma$ and move to state $q$. Thus we can indeed push several symbols at once, and we always pop one. With this single instruction we can also pop a symbol (by not pushing anything back) or keep the stack unchanged (by pushing back what has been popped).

In the Sipser book the formalism is different for instance.