Egypt will eventually become a Coptic Kingdom. Conquering Egypt essentially means following the Nile where 90%+ of Egyptians lived, the Mongol invasion as always will devastate the Nile Basin which would lead to massive deaths, destruction of farms, essential canals and irrigation, settlements, killing of livestock etc... Pretty much what's happened in Persia that set back the region for centuries, but concentrated around the densely populated Nile where it's effect is even more horrendous.
The Mongols will convert to Coptic Christianity, expand to christian Nubia and Axum, control the red sea trade and later on deposed by a native rebellion...
Destruction caused by the Mongolian invasions depended upon the resistance: usually they were offering an option of a peaceful surrender and started doing all their favorite things (burning, looting, etc.) only after being refused. As far as the region is involved, King of a Lesser Armenia declared himself to be their vassal and participated in their operations with his troops. Taking into an account Hulagu's pro-Christian sympathies and his dislike of the Muslims, it can be assumed that, if the things are favorable to him he'd proceed with more or less the same policies and find Coptic cooperation in Egypt. Which of course assumes that he is not looting and destroying too much (hardly a good scenario for gaining their sympathies).
Taking into an account that majority of the region's population were Muslims, it is rather questionable if he was going to see his task as a complete annihilation of his future subjects, especially when a big part of his army are Muslims. The stories about the Persians not being able to get back after destruction of the irrigation system built, IIRC, some time BC does not speak well about the local population (aren't they still complaining?).
Under any scenario the Mongols would not be able to stay in the area because neither Syria nor Egypt could support big numbers of horses and cattle. Neither would they "convert to Coptic Christianity": their Christians were of the Nestorian creed.