You asked for it, here it is. Read it and compare [thread=32411]with the new, reworked TL[/thread] to see how and how much my TL has changed. I'm gonna post it once per century (at least as long as things don't happen too fast).
So here's the 1200s:
"Alexander Newskij and the Hanse"
1200: Ghengiz Khan defeats the Tatars, but dies shortly after. There are his sons Chagatai, Jochi, Tului, Ögädäi; the latter one is gifted, but too young.
1204: Fourth Crusade. The crusaders defeat the Byzantine Empire, plunder Constantinople and found the Latin Empire, which only has one quarter of the Byzantine lands, though. Other parts go to Venice or become independent. One of the new states is the empire of Nicaea, the ruler of wants to kick out the Western invaders. In 1261 they'll be successful, taking Constantinople back.
1240: The Russian prince (there are no czars around yet) Alexander defeats the Swedes at the Newa, acquiring the honorable Newskij. Now Russia has a place at the Baltic Sea. In the following year, he founds near the place of his victory the city of Alexandersburg. Soon ships of the new Nothern German trade alliance, the Hansa, go to the new port, trading Russian pelts and food against the products of German artisans - amongst other things, weapons too. The Hansa controlls all of Russia's external trade and doesn't want anyone to break their monopoly, which ticks off some Russians.
1242: Alexander proves again what he can, defeating the Livonian knights on the frozen Lake Peipus.
IOTL that's all he could do, but ITTL he isn't controlled by the Mongols. He wants to unite the Russian princedoms, which he achieves after 20 years of war.
1260-62: Alexander makes another war, this time against the Kumans, living south of Russia at the shores of the Black Sea. He wins, conquers Odessa and sends a ship to the new Byzantine Empire. Delegates from Constantinople visit the Russian capital of Vladimir and are received by the old grandduke. By them, the Russians learn about how Constantinople was captued by the Franks (=Western Europeans).
After 1263: Alexander's successors aren't that interested in Western Europe.
They don't have a clear line for what to do in politics: One of them helps the Byzantines to win back lost areas, others conquer the Kumans and advance to the Urals in the East. 100 years after, Russia owns all of the eastern half of Europe.
1291: Akko, the last city of the crusaders in Palestine, is taken by Aultan el-Aschraf Chalil. The time of crusades is over.