Does the German surname "Untermeyer" (as in the famous poet Louis Untermeyer) mean a subordinate?

Partially yes. A "Meier" (of which "Meyer" and "Maier" are alternate spellings) was a bailiff or an administrator of some plot of land, who himself wielded some often considerable power. "Unter" means "under" (as a subordinate), so that was a lesser / subordinate Meier (but still a low-level manager, so not entirely a subordinate).

https://en.wikipedia.org/wiki/Meier

In German, "Unter" and the opposite "Ober" are common modifiers to office / rank names, such as in "Oberleutnant" (first lieutenant) or "Unterfeldwebel" (junior sergeant)