A language which is neither r.e. nor co-r.e
Recall that a language is r.e. iff its complement is co-r.e. It follows that a language is neither r.e. nor co-r.e. iff its complement is neither r.e nor co-r.e., just like a language is recursive iff its complement is.

The language $L_{\exists}$ is recursively enumerable, though not recursive. To enumerate it, simulate all Turing machines on all possible inputs, and output a Turing machine whenever it halts on some inputs. To see that it's not recursive, reduce the halting problem to it.

Your other example $L_{\forall}$, usually known as TOT, is indeed neither r.e. nor co-r.e. In fact, it is $\Pi_2$-complete. You can prove that it is neither r.e. nor co-r.e. directly, by finding appropriate reductions from the halting problem.