The Egyptians would have a problem with non-Muslim rulers, but what I meant was there is no way to get a Mongol army into Egypt. It would turn into a puddle of horse-goo crossing the Sinai.
An initial invasion might have no compunctions about ravaging and consuming everything to fuel their army, but that's not going to work well in an extended campaign to reduce Western Europe - impossible, I might say.
Persia allied with the Avars pulled the same thing with the Byzantines. By the nadir of Byzantine fortunes, little was left of the empire BUT the city, yet Heraklios managed to crush the Persians and get it all back. Without destroying the administrative center of the empire, you're still stuck with trying to cross the Bosphorus, reduce the fortified Byzantine cities in Asia, which largely have sea supply and are very difficult to deal with, and no real access to the European parts of the empire, which by this period are the core, plus all the islands.
As for lasting Mongol states, true, they were rare, but the Golden Horde did pretty well and lasted as the Crimean Khanate until the late 18th c, so that's a good 500 years. I don't think they will necessarily be assimilated everywhere, just where there are sophisticated cultures with large and dense populations. Much of Eastern Europe will assimilate into their culture - look at Seljuk/Ottoman Anatolia and the Crimea.