Push Down automata:- Basic doubt
In general we do not assume that push-down automata have an end-of-string (or end-of-input) marker. 

PDA have two modes of acceptance, with final state and with empty stack. Even when the automata are written for final state acceptance it is helpful in the design to check that the stack is empty. That ensures no data is unprocessed.

The PDA on wikipedia for the language $\{0^{n}1^{n}\mid n\geq 0\}$ has transitions: $(p,0,Z,p,AZ)$,
 $(p,0,A,p,AA)$,
 $(p,\epsilon ,Z,q,Z)$,
 $(p,\epsilon ,A,q,A)$,
 $(q,1,A,q,\epsilon )$, and
 $(q,\epsilon ,Z,r,Z)$. Initial state $p$ and final state $r$. This indeed is of the form you describe.

The same language can be accepted without $\epsilon$-transitions at all.

$(p,0,Z,q,Z)$, $(q,0,Z,q,AZ)$, $(q,0,A,q,AA)$, $(q,1,A,r,\epsilon)$, $(q,1,Z,s,\epsilon)$,  $(r,1,A,r,\epsilon)$, $(r,1,Z,s,\epsilon)$. Initial state $p$ and final states $p$ [to accept $\epsilon$] and $s$.