I agree with @Mark Robinson. The introduction of do in questions (What do you think?) and in reference to the past tense (Did he swim?) is a relatively modern. Because the usage without the grammaticalized do is more archaic, it also sounds more formal to our ears, but was not originally formal. It was just English.

Now, we use it to heighten the question's intensity, or poetic purposes, or just for verbal flare. It is perfectly comprehensible to native speakers, so the only concern in using it is whether its oddness will be appreciated as you hope in any particular situation.

A sentence with "do" is more common, and almost never sounds odd.

A sentence like "What say you?" is mostly used for special effect, perhaps in a poetic environment.

You can use those phrases, however they aren't commonly used for most things. They are older ways of saying stuff.

They aren't used much today except for a few cases as you mentioned. For instance, “what brings the mailman?” or “what says the man?” as well as “what say you?” which is occasionally used on legal dramas.