Getting away from the Jewish question, one TL in which the Crusaders could keep Jerusalem is one in which the Seljuks never make it to Anatolia, being stopped by a much stronger Abbasid Caliphate. This could possibly come about by them holding onto their territories in Iraq, and therefore keeping the Seljuks in Persia, possibly as a vassal state
If the Seljuks never make it to Anatolia, chances are there will be no Crusades. The Crusades were due, in no small part, to the Byzantine emperor getting tossed out of Anatolia after the battle of Manzikert. And if the westerners turned up in large numbers when the Byzantines were still strong in Anatolia, then the chances are that the Byzantines would oppose the Crusaders. Crusaders and Byzantines got along pretty poorly, and the Byzantines tolerated the Latins only because they were useful against the Turks.
Secondly, by mid-late ninth century, the Abbasid Caliphate had ceased to exist. To keep the Turks in Persia/Khorasan, you need to butterfly away the native Iranian kingdoms that rose in the ninth century with the weakness of the Abbasid caliphate. A better idea would be for the Samanids/Buyids (who still retained a lot of vitality) to hold their own against the Turks and keep them out of Persia. Abbasids and the Turks got on much better than the Iranian dynasties got on with the Caliph at Baghdad.
Then, they could use their stronger positions to reconquer the lands in Syria held by the Samanids,
I think you mean the Fatimids.
and start encroaching into Byzantine territory, which would then cause the Byzantines to ask for aid.
This is a bit far fetched. A stronger Abbasid Caliphate would be tempted to regain its original eastern territories. However, even if they turned west against the Byzantines, they would not be able to hold territory in Anatolia easily. The Byzantines would regain their territory much more easily against the larger and more easily distracted Abbasids. The Seljuks were effective because they set up their kingdom in Anatolia, instead of Baghdad or Damascus.
A much better coordinated Crusade then happens, securing the Levant. Some states are set up, protected by a mix of Crusaders and Byzantine forces and settled by Christians.
This is very unlikely. If there are very successful Crusades, the Byzantines and the Latins would squabble and if the Byzantines proved successful, the Crusaders would sulk and may be return home, ignoring Byzantine interests in the region. If the Latins proved more successful in the squabbles, the Byzantines would be tempted to choke off supplies and reinforcements, to bring the Crusaders into a more cooperative frame of mind.
To protect themselves from Muslim aggression they combine into one Principality, paying tribute to Constantinople in return for Protection.
There was never a Christian vs Muslim war for long. The battles were essentially free for all affairs. If the Crusades were successful, the Byzantines and Crusaders would squabble. Period. The only way to avoid this is to re-unite the Greek and Latin churches. Otherwise, rabble rousing priests from both sides would be tempted to fish in troubled waters and set their minions on each other. This is ignoring the fact that both Crusaders and Byzantines treated the local Christians pretty shabbily. Syrian, Armenian, and Coptic churches all got kicked in the teeth by the `only true faith' wielding Crusaders and Byzantines, both of whom, if successful, would be even less inclined to negotiate with the locals.
The loss of Jerusalem causes the Caliphate to break up into feuding rulers, which allows the Byzantines and Crusaders to expand further.
They were already operating at the far end of their logistical limit. How much further can they expand, with the pathetic number of troops they have at their command, before collapsing totally under the weight of their own conquests?
Then the Mongols turn up, chewing up the remains of the Caliphate and causing the Crusader states to become a vassal of the Byzantine empire in return for their help in fighting off the Mongols.
Or something. This time period is not my speciality.
If the Mongols find the Christians uncooperative and unwilling to bow to them, they would be razed to the ground along with any other stiff necked lot. Mongols treated all opponents, Christian or Muslim, with their usual ruthlessness. If the Christians refused to bow to the Mongols, they would be crushed as well.