Nauta and poeta are both members of a small set of profession words which are masculine despite being in the first declension (-a) in Latin. My textbook provided a comprehensive (again, small) list and just suggested they should all be memorized; the only other one I remember now is agricola, farmer.

Manus, hand, is one of an even smaller set of feminine fourth declension nouns. I believe this oddity has also been preserved in Spanish as "la mano".

This makes me wonder whether the Spanish words for trees tend to be feminine while having masculine-looking forms; that is regular in Latin.

most nouns ending in -ma (but not all) are masculine. Basically the ones with Greek roots. I was taught -ma is masculine like your mom.