What if Batu Khan invaded the Byzantine Empire after taking on Russia, instead of pushing into Hungary and Austria as in OTL? Why did they leave Constantinople alone anyways?
Simple: the first reason was politics; the Hungarian king Bela IV had allowed Cuman refugees to settle in his lands, with which he incurred the wrath of the Mongols, because the Mongols had defeated the Cumans, and they regarded Bela's cooperation with the Cumans as an act of defiance.
Besides, the Hungarian kingdom was situated in the Panonnian Plain - a huge plain that mainly consists of grasslands - and the Mongols were looking for new pastures to conquer, so the Mongols would have been attracted to Hungary anyway.
The second reason is the opportunism that was so typical for the Mongols; it was much easier for them to loot the Hungarian and Bulgarian lands and cities, and Constantinople had lost much of its wealth and glory due to the Fourth Crusade.
And both Hungary and Bulgaria were simply closer to the Batu's territories than Constantinople.
The third reason; even tough Constantinople lost a good bit of it's wealth, its formidable defenses were still largely intact.
And what's more; Constantinople was surrounded on three sides by water, which makes it impossible to lay siege to the city and attempt to starve it into submission as long as you don't have naval supremacy as well - and to put it simple: the Mongols sucked at naval warfare, and none of the western Khanates had a fleet.
The Mongols had a habit of avoiding enemies that they felt were either too tough or powerful, or just not worth the effort of conquering - and I'm pretty sure that this is why they pretty much ignored Constantinople in OTL.
...and the fourth reason is that the Byzantines understood fully well what kind of a (potential) threat the Mongols were, and the Byzantines did pretty much anything to avoid any confrontation with the Mongols.
What if the Mongols sized Constantinople and made it a Khanate centuries before the Ottomans captured it?
OTOH, Constantinople (or the remnants of it) would be much more easier for the Ottomans to capture ITTL. Considering that Ottoman was a Balkan Power before Constantinople was captured in 1453.
If the Mongols capture Constantinople, I'm pretty sure that the Ottomans will be butterflied away alltogether - the emergence and rise of the Ottomans (along with the other independant Anatolian Turkish principalities, for that matter) happened because of a very specific set of circumstances.
The early Ottoman state emerged when there was basically a power vacuum in Anatolia; the Mongol Ik-Khanate had lost much of its power in Anatolia as a result of continueous conflicts with both the Mamluks and the Golden Horde, and the Byzantine Empire was suffering from one of the worst cases of mismanagement in history thanks to emperor Andronicus II, who was coincidentally also one of the longest ruling Byzantine emperors (he ruled from 1272 to 1328).
If the Mongols under Batu capture Constantinople, and actually make a serious attempt to conquer the region, then this will drastically influence the political situation in both Eastern Europe, as well as the Middle East.
First of all, if Batu and his successors focus on conquering the Balkans, then this will mean that the political centre of gravity of the Golden Horde will move westwards - the Mongol leadership will most propably end up settling (at least temporarily) in the Panonnian Plain (which would be an excellent base of operations for the Mongols), from which they would organize raids and campaigns in both the Balkans as well as Central Europe.
This is an important difference with OTL, because IOTL, the Golden Horde began to focus on the Il-Khanate when Berke Khan rose to power.
The Golden Horde already had a border conflict with the Il-Khanate over the Caucasus and what's now Iranian Azerbaijan (Batu and his successors claimed that this region was allotted to them when Genghis Khan's empire was divided, but Hulegu Khan claimed this region as well), but Berke Khan went even further; he declared war againest the Il-Khanate, and even allied himself with the Mamluks of Egypt againest the Il-Khanate.
If the Golden Horde becomes focused on conquering the Balkans (and I'm fairly sure that they wouldn't attempt to take Constantinople unless they were already bent on conquering the Balkans), then it is quite possible that the border conflict with the Il-Khanate will become less important, and if Berke Khan is butterflied away, then there most propably won't be a war between the Golden Horde and the Il-Khanate.
And this leaves the Il-Khanate free to take on the Mamluks.
As a result, the Il-Khanate isn't weakened by war on two fronts againest the Golden Horde and the Mamluks, and thus manages to remain stronger in Anatolia.
And Nicaea, which most propably remains confined to western Anatolia due to the strong Mongol presence in the Balkans ITTL, will remain stronger too, as IOTL, Nicaea's wealth and resources were largely drained by rebuilding Constantinople and wars againest the Serbs and Bulgarians.
...and if the Il-Khanate remains strong in Anatolia (ergo; brutally destroys every uppity Turkish principality and/or warlord that refuses to submit), and Nicaea remains strong and stable as well (instead of being plagued by overtaxation, mismanagement and a lack of a proper army to protect the place), then the power vacuum and other conditions that allowed the early Ottoman state to emerge simply won't exist.
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As for what would happen in general if the Mongols captured Constantinople; like I pointed out earlier, a Mongol conquest of Constantinople is only likely if Batu and his successors keep focusing on Eastern Europe in general and the Balkans in particular.
Therefore, a Mongol Constantinople would also mean a much stronger Mongol presence in Europe, with the westernmost centre of power (of the Golden Horde) being somewhere in the Panonnian Plain.
Constantinople would still remain a relatively important city, but it would not be a capital of anything more than a province until the Golden Horde falls apart, and some local dynasty manages to take control of Constantinople and build a new state from there.
But until then, the more interesting events in Eastern Europe will take place in whatever city in the Panonnian Plain the Mongols make their capital - once the area starts to recover from the conquest, that capital will start attracting merciants, artists, craftsmen and scholars from not just Eastern Europe, but Central Asia and the Far East as well.
That should lead to some rather interesting cultural cross-fertilisation...
At the court of Hulegu Khan of the Il-Khanate, Chinese astronomers were working side by side with Persian Muslim astronomers, with interesting results - something similar could easily happen in this Mongol state centered in the Panonnian Plain...