Is the empty string and some words of even length are elements of this set?
Yes, since "the first, the middle, and the last characters of $w$ are identical" is not really well defined here (so there might be ambiguities like is $\epsilon\in L$?)

Probably it wont really matter for the question if you assume $\epsilon\in L$ or $\epsilon \notin L$ (other than some small change in the PDA or CFG) But if you really want to make sure everything is ok, then write (in the answer for the question) that there are ambiguities and this is how you interpeted them.