Why would Amalric "auction" Egypt off when he could keep it himself? Its still a very rich country with plenty of resources Amalric could use. And why give power to the Copts? The OTL Kingdom of Jerusalem wasn't exactly nice to the Oriental Orthodox (which is what Copts are) within its borders-in fact, they acted like such dicks to them that some Oriental Orthodox contributed troops to Saladin to fight the crusaders.
Personally, I've always thought that crusaders would do something similar to what they did in the Levant-eliminate the local nobility and replace them. Egypt would become a country with a native underclass-majority Arabic-speaking Islamic in the north, majority Coptic in the south-and an upperclass made up of French, Italian, and German-speaking nobles living in estates all up and down the Nile (as Egypt has a lot more good farmland than Palestine, and Amalric probably won't want to abandon Jerusalem, I expect much of Egypt would simply be offered to any Catholic noblemen willing to come over-something that might prove attractive for third and fourth sons). Venice and Genoa would likely receive trading quarters in Alexandria, Cairo, and maybe some other places, and come to dominate international trade, and maybe even trade along the Nile as well (is it navigable for their galleys? honestly don't know)
It will probably be somewhat weaker than the Abbasids were, but put a good deal stronger than the OTL crusader states. There aren't any real Islamic powers close by, and I could see it lasting for several centuries, until some large Muslim state emerges in Anatolia, North Africa, or Persia/Iraq that could take it on. Heck, wasn't most of Sudan still run by Coptic Nubian states at this point? Its fun to imagine someone uniting them and doing in Crusader Egypt, though that's probably the least likely scenario.
Might have some interesting effects-Egypt will probably have a much larger Coptic minority, and maybe even a surviving Coptic language, since Latin rule will put Islamization and Arabization on hold while it lasts. Europe would have even more contact with Middle Eastern science and philosophy, and I can even see the Italians using Egypt as a base to trade with East Africa or India. Heck, it might be the fall of Egypt that causes TTL's Portuguese to try and go round Africa.
And the Fourth Crusade is likely butterflied, though the Latin Kingdom of Egypt, allied with some Italian states, could cause problems for Byzantium if it wanted to.