Does PDA epsilon transition only transition when there's no more input?
A PDA is always allowed to take $\epsilon$ transitions, just like an NFA. More generally, the PDA is not allowed to peek at anything beyond the current letter. It cannot tell whether it has finished reading the word or not. Whether it can take a transition or not depends only on the contents of the stack and, in the case of a transition "swallowing" an input symbol, on the next symbol in the input (if any).