I'd also be interested in hearing how it played out with other Christian groups like the Maronite Catholics in the Levant (debate about Assyrian, Arab or Phoenician ancestry aside) or Assyrian Christians spread throughout the Levant, Anatolia and Mesopotamia.

Great question though.

Since it has been nearly a day since this question was asked, I will try to answer as correct as I could. Most of my sources are Coptic sources, I basically translated from the Arabic references. All in all, the crusades weren't good news to the Coptics. We will look at this from 3 Perspectives:

The Catholics considered them heretics since they disagree upon the nature of Jesus Christ. Also, the European political leaders' primary goal were to have what will centuries later be identified as colonies in the east and control important trading routes. The Pope in Rome wanted to become the head of the Christian church and capturing the city (Jerusalem) mostly associated with Christianity would achieve just that. The existence of Coptics for them were problematic since they heavily disagree with them. So the crusaders treated them quite harshly.

The Muslims on the other hand didn't really have a good grasp on the exact difference between Catholics and Coptics. The term used to describe the crusades forces were Elsalybeen "Followers of the Cross" and virtually since both churches revered the sign of the Cross. Muslims considered them the same and initially mistrusted them. This confusion led to some contradictory decisions. For example, Saladin the Sultan of Egypt and Syria fired all coptic employees in his government then rehired them again. There were rumors that the Mar Mark Cathedral was supporting the crusaders, who will use it as a fortification when they capture Alexandria so Saladin demolished it. After he recaptured Jerusalem he gave the coptics a monastery in Jerusalem as a gift and appointed a Coptic Minster Ibn Sharki.

The Coptic take was that they had deep differences with the Catholics and their situation was quite dangerous since most of their followers live under Muslim leaders. During the Ayyubid dynasty their living conditions were comparatively good. They filled important jobs and were near the decision makers so there was no incentive to switch sides. The Catholics considered the Coptics' support for Muslim as treason, so they further mistreated any coptic subjects they came across.

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Sources:

1- "Egypt during the Middle Ages" by Mahmoud Elhouery

2- "Coptics and Muslims since the Arabic Conquest" by Jack Tager

3- "History of Eastern Christianity" by Aziz Sourial

Edit: wording and Grammar