I've posted this some time ago, should still be valid:
1) maybe this you know already - you can go to your liege, if you're a lord to someone, and ask him for a couple of months of leave to "help our brothers in the holy lands". Then you won't count any longer as a member of the faction but you are now a member of the Regnum Hierosolymitanum, i.e. the relations with your current faction is set to 0 as well as the relation with your faction's enemies instead of that you will get a relation boost to 100 with Patrimonium Sancti (as well as some major relation boost with the pope), some +12 with Reg. Hiero. (I think it was something like +12) and -100 with the Mamlukes either/or -40 with Ilkhanate respectively if Regnum Hierosolymitanum is at war with them. On returning from the crusade and re-joining your original faction (by talking to the king "My lord I've returned and am ready to join you again" or something along the line) you will get +7 relation boost with the king of that faction and some +12 relation (not sure) with your original faction and of course the negative relation with Mamlukes and other enemies of Reg. Hiero. will be set to 0 again. If you are on a crusade and conquer a castle/town, request it for yourself and get it awarded it will belong to you but will still be a part of Regnum Hierosolymitanum.
This can be some kind of a problem, for instance: in one of my games I was a lord in Denmark but I had re-conquered Acre while crusading and got it awarded as my fief. So far so good, but then the Mamlukes started to besiege Acre (later the Ilkhanate and the Roman Empire laid siege to it too, once or twice) and every time I not only had to travel all the way down to Acre to defend the city but also had to ask my liege for leave beforehand every time, as I else was not able to join the fight for Acre. In this way I managed to hold Acre for nearly 2 years but finally gave up on it, which was particularly sad because by that time it was the last town/fortress of Reg. Hiero.
2) if you're a lord/king you can go to the pope and tell him that you wish to start/finance a crusade (it costs 100,000 dinars!), then you have to choose against whom you are going to crusade (which faction) and pay the money. It will then appear a message, that your faction has declared war on the other faction. After that there will appear a couple of hundred troops somewhere in your faction (I think near the main fief, i.e. castle/town, of the faction's marshal). There the crusaders will wait until they have accumulated to ~1000 men (will last a few days), then they will start their journey and will lay siege to some castle of that faction you've picked earlier by talking to the pope (you cannot choose or tell them which settlement to conquer). The marshal will automatically get that fief which the crusaders conquered in the name of your faction, so if you're the marshal you will get the fief. Sometimes crusader armies are stopped by marshalled enemy armies. You cannot interact with the crusader army but you can of course attack them if you're a pagan/muslim character. I think you always can attack a crusader army independent from your religion or faction and I think it is even possible to attack your own crusader army
The AI can crusade all on their own, as you've surely realised yourself (every christian nation can pull off a crusade against any non-christian/pagan nation). It happened for instance in my current game where I am the marshal of HRE. So, one day there appeared a message that HRE declared war on Lithuania and I thought "What, why the hell...?!" - then I realised what it was, after suddenly some 400 crusaders appeared near my main fief which is Wroclaw. A few days later they had accumulated to something like 1050 men and went off for their aim in Lithuania, conquered a castle there which I as the marshal automatically received as my own fief. I've never seen any crusader army besieging a town but this could be due to the towns garrisons being way stronger, as a rule, than those of some castles. They seem to always pick some castle which is the farthest away from the crusading faction and/or the weakest castle. Although, in another game I was Emir Suleyman ibn Al-Jazira of Marinid Sultanat and had to fight off two crusades against our faction, and especially me, as it seemed that the AI somehow favoured my castle, although it had ~700 men in garrison (twice as strong in numbers and quality of troops as any other castle garrisons, even stronger than some town garrisons) and was at that time right in the middle of the Marinid empire, Al-Jazira castle. But I didn't and do not complain because I loved to fight the crusaders off.
