Whoa. I'm not sure if the Russians would have been *that* drastic, but then again Alexander was an odd bird (though not as weird as Paul). I guess that's entirely possible; the Petersburg court moves back and forth between that city and Moscow a lot already as it is; why not a third "capital"?
Throughout 19th and early 20th century, emperors and ministers kept looking for new capitals; St. Petersburg was getting very uncomfortable, whereas a capital in Warsaw or Kiev (two very popular choices) would have both helped with the European/southern focus respectively and, it was thought, would've reconciled the local uppity minorities, assimilating them into the empire and turning their fervour to work in its favour. Constantinople wasn't all that good (if only because it wasn't conquered yet; however, it would've also been very difficult to defend and rather detached from the main empire), but made up for it all by being Constantinople.
There is a new such craze nowadays, btw. St. Petersburg, Samara, Ekaterinburg, Novosibirsk and Vladivostok all strive for the status of the next capital. Obvious superiority of Ekaterinburg aside (

), it might be nice to see how Vladivostok would work out.
Who would get Belgium? The Austrians don't want it at all anymore, and the Dutch Republic isn't in the British good books, while the Oranje don't have enough support to even get the Seven Provinces as of 1805-6.
Make it independent and put a Hannoverian on the throne. Or hell, just add it to the personal union.
The industrial might of Britain and Belgium united is the stuff of nightmares, though.
I don't see much of a remedy at all outside of giving them basically all of northern Italy.
Well, why not? I mean, everybody else with serious influence in this place (a.k.a. France) isn't in position to do anything about it, and I don't think Prussia, Russia or Britain would be really opposed to it; they all have their own backyards to deal with, why not throw the Habsburgs a bone as well?
What if Tokhataymesh won the war against Timur and preserved Golden Horde's power?
The reason he lost the war was that there was no real power to preserve. The Golden Horde was already disintegrating; even had Tokhtamysh pulled off a victory (which
is interesting, mind you, as the only way to get a lasting victory against Timur would probably be to neutralise him, one way or another; otherwise he'll just go back, raise another army and destroy everything in his wrath; but if he's dead or captured, then the Middle East will probably fall into infighting again, while the Ottomans would not be pushed decades behind in their progress), it would've imploded soon enough.
What if the Malmukes had lost at Ain Jalut, or the Il-Khanate was not torn apart by civil war and the Khan was able to revenge the defeat and advance into Egypt and further into Turkey and such?
It was not "torn apart by civil war"; not yet, that is. Abaqa, the son of Hulegu, did try to get revenge, but once again failed miserably. The Mamlukes were fighting in their own home turf, while the Mongols had already ran out of steam and lost their main advantages.
Recognising this, the next ruler converted to Islam, allied with the Mamlukes and began transforming the Il-Khanate into a proper sultanate. That could've worked out pretty well, actually, but no, Buddhists (namely, Arghun, Abaqa's son) just had to go in and ruin that, and screw over the Il-Khanate's potential. Bah.