To deal with the second question first: There was no such thing as midshipmen in the 1300s and 1400s, those would have been much later inventions.

For the first question: In the English and French navies, dried meat and stockfish, as well as bread (biscuit) and cured or dried vegetables would have been standard fare. Ships would carry fresh water, but also beer and wine, as well as regional foods; N.A.M. Rodger's The Safeguard of the Sea has a throwaway reference I've been meaning to look up about the "astonishing quantities" of mustard carried on Burgundian ships.

I don't know anything in particular about the Spanish or Portuguese caravels, but the same principle would likely apply: foods that can be dried or cured, with fresh stuff in port.