It's odd that AoE always portrays the Turks as the "gun" civilization. In their own time they were not known for any extraordinary skill with firearms or artillery.
Depends which Turks. The original Turks, Seljuks, were a steppe people, similar to the earilier Huns and the later Mongols. The later Turks, Ottomans, did indeed start utilizing artillery pretty heavily at some point, that's how they took down the walls of Constatinople in 1453. Not sure about handguns, but since they fought against the Europeans quite extensively and effectively throughout the 1400s/1500s/1600s, I imagine they probably had to have had at least some handguns/arquibusiers/muskets.
They used only as many guns as their contemporaries, and they weren't pioneers in the field, which is why I don't think it makes sense as their national theme. Their artillery experts were foreign christians, maybe because christians were good at casting brass for church bells already. At Constantinople, the giant cannon that's their national unit was cast and crewed by a christian mercenary. The turks didn't even know the correct way to bring down the walls with their cannons until a venetian diplomat casually explained that they were doing it wrong. Yes, the turks had guns, but so did everyone they fought against at the time, including the constaninople defenders. The turk's firearm technology was never ahead of contemporary europeans. What the ottoman turk military was known for was their skill as sailors and as cavalry, which would be a unique combination of strengths in the game. If you look at what europeans were writing about turks at their zenith, their concerns were 1. how to win a battle against 200,000 horsemen, and 2. how to deal with the turkish corsairs who were ravaging as far as spain and southern france. The turk's outmanuevering of the venetian and genoese navy at constantinople contributed much more to the victory than their giant meme gun, which could have lost them the siege when it burst and nearly killed sultan mehmed.