Catholic factions are good fun because of the quality of your heavy cav and foot units, and because of the added complication of dealing with the Pope....Orthodox factions are good because of the variety of mounted archers...I will confess that even when playing as a Catholic faction I use Turcopoles/mtd Xbows whenever possible...I will confess I have never tried my hand at a muslim faction (though I have often been tempted to try the Turks/Seljuks).

Faourite factions....hmmm....don't know. each that I have played has had its own appeal. Some of my personal faves:

Sicily - location (access to trade etc), ILI....
Denmark - location again, vikings, longboats
England - billmen. Longbows are okay, but I far prefer arbs
Aragon - location, jinetes...didn't go far enough to get Lancers
Byzantines - some great units, and a "meat in the sandwich" position
Serbs (XL 3.0) - threatening start pos, access to all manner of mounted missile units (HA's, mtd Xbows, Carska Garda, Serb Cav...)

Decisions, decisions!