The Mongols could've all been die-hard Roman Catholics, it wouldn't have mattered much. They had alot of Nestorian Christians, they were generally friendly to and tolerant of Christians, but their world was pretty much divided between those who have submitted and those who still needed to be conquered. It might be fair to say that they didn't much understand the concept of 'ally'. While temporary military cooperation in order to achieve their interests wasn't out of the question, such an arrangement would last only as necessary. Once it was Europe's turn in their sights (if they ever got around to it), such past 'friendship' would likely be of no avail. Their goal until the fragmentation of the empire was largely consistent; Global conqest.
If the Mongols had won Ain Jault and defeated the Mamlukes, every Crusader state (except Jerusalem, where some alternative arrangement might be made to quiet the West) would be forced to become Mongol vassals. They might maintain local autonomy and their existing Latin leaders, but they would be tribute paying subjects of the greater Mongol empire. And noone in Western Europe was likely to commit the resources to change that reality.
As for Ain Jalut specifically, I think it's highly relevant only by applying centuries of hindsight. Yes it essentially marked the extent of the Mongol empire's Western push, but I think its a stretch to call it the cause of that halt. The Mongols were able to field armies surpassing 100,000 men at that time. Of course by this time most were subject people armies, but they still had a strong core of steppe troops and there was really no match out there in skill or numbers. They fielded only about 20,000 at Ain Jalut because Haluga had to return east with most of his army when the Big Khan died for elections.
Its entirely likely that once Haluga returned he could've overrun the Mamlukes with massive numbers and rendered Ain Jalut a minor speedbump in history. What stopped him was the fragmentation of the mongol empire and his subsequent war with the Golden Horde.