I understand that, with a few notable exceptions, knights were generally high born, but what about other professional soldiers? Men-at-arms, crossbowmen, archers, pikemen, etc?

There was something of a class gradient among soldiers. Men-at-arms were usually of relatively high birth – the younger sons of knights or at the very least from the gentry or esquiry – specialists like Swiss pikemen or Genoese crossbowmen tended to be somewhere below them but still not among the poorest, but archers were more likely to come from the peasantry b/c of the bow’s use for hunting in rural areas.