I've put some thought into this, and just want to make sure I'm on track, or if I need to be corrected. Basically, my answer is this: Yes, you need to always specify a domain when formalizing into predicate logic, an unrestricted domain results in paradoxes. I guess in general I'm not exactly sure what the basis is for demanding that you specify a domain in predicate logic, other than that is a requirement of the completeness proofs or other kinds of proofs for it.
On the other hand, an unrestricted domain does seem harmless at first, for when you say "All men are mortal", the first predicate "man" does seem to restrict the domain for you. Even so, it might allow other "categories" of "man" that you didn't intend: fictional men, men in the future, toy soldier men, etc. So the option here seems to be that either you restrict the domain to an ontological/semantic category, or the category should be contained in the predicate. As an example of the latter, it could be implied in the predicate "man" above that the category is restricted to contemporary, flesh-and-blood, and real human beings. Is the choice on whether the category should be determined by the domain or contained in the predicate arbitrary or is there a good logical reason for preferring one over the other?
Since I was thinking about categories, and wondering about domains in the context of "category mistakes, I found the SEP article on "Categories" and there they (referencing someone named Thomasson) linked domains with the difficulty of Russell's Paradox. So, first, does an unrestricted domain cause Russell's paradox? Second, is there some sort of philosophical connection between Russell's paradox and what are known as ontological or semantic categories? By ontological categories, I mean the systems of Aristotle, Kant, et al. By semantic categories I mean the analysis of category mistakes by Ryle, Sommers, et al.