Is $AlwaysHalt$ recursively enumerable?
Consider the language $ALL_{TM}$ (sometimes also called $L_{\Sigma^*}$)
$$ALL_{TM}=\{\langle M \rangle: M \text{ is a TM and } L(M)=\Sigma^*\}$$

It is well known that $ALL_{TM}$ is not RE (nor co-RE). See, e.g., this question (and search for other related questions on this site).

From this fact, it is easy to see that AlwaysHalt cannot be RE: a reduction
$$ALL_{TM} \le AlwaysHalt$$ is rather trivial.