Don't know about the stone walls, but Portugal can train Jinettes from the Plaza del Toro, or from a racetrack if they capture one. Are you sure you don't have one of these in the city in question?

Portugal can't train anything from the military academy, I'm not sure why they get them. It gives some very nice traits and ancillaries though, so it's worth building. Plus many of them help gunpowder or artillery, so I suspect it's to give Portugal a little bit more of an edge since they're such a gunpowder-heavy faction.

PAs are very nice units indeed, not quite as deadly as Jannisary or Cossack muskets but much cheaper to get and I think a little tougher and more capable in melee. The thing which always annoys me about gunners is that the AI always seems to target them with its archers and towers which pick them off depressingly easily, so the extra armour is certainly very welcome.