Yes, even European plate was known to overheat to the point where the wearer's heart gave out. That's why the Crusaders (when they managed to get to the Holy Land) never went up against similar military forces - the Saracens lived there, and knew about the practically indestructible tanks that were European heavy cavalry, but never equipped their troops with plate. They couldn't. They relied on chain mail, light horses, and mobility rather than the sheer irresistible force that was a plate-armoured knight. Bear in mind, these weren't desert nomads, these were the most advanced civilisation this side of China.
So a medieval Egypt would likely develop chain mail to compete with its neighbours. Elite warriors, with suitable magic on their armour, could be expected to wear plate, but this is expensive and thus rare. It means you see a gang of normal chain-mail soldiers charging towards you, with this huge ibis-headed guy at their head clad in plates of metal - probably inlaid or plated with precious metals, gems, lapis lazuli etc; the Egyptians liked that sort of thing.
Chariots I'm not so sure about. They are a fairly iconic Egyptian contrivance, but there's probably a tactical reason they're not seen in the Medieval period. They probably figured out that a man on horseback (with stirrups and advanced armour and training to use a bow) is just as mobile and not as expensive to repair. However, don't let that stop you throwing in chariots filled with archers or hulking great kapesh wielders lopping off heads left and right, because they're cool.
