So the Mongols just say "F*ck it, theres nothing there except Snow, Bears, Peasants, and Snow." And don't invade Russia, or they just lose the invasion or something. But they still invade the Middle East and such.
Why wouldn't they? Besides the fact that Mongolia itself and much of the early conquests were snow, yaks, herders, and snow, Russia was actually a lot more. There was a lot there: important trading routes, a great many urban settlements, and an advanced civilisation (Kievan Rus, which had politically fragmented before the Mongols but still left a cultural legacy, had been one of the most literate places in Miedieval Europe). The Mongols certainly had every motivation to invade, vassalise, and tax the principalities (which was all the "Tartar yoke" ever actually was). As for their loosing, sorry, they're the
Mongols! The Russians never stood a chance.
A lot a people say that if the Mongols didnt attack Russia, then it wouldn't be the nation it is, so that would mean there would be Russian States like Novogorod and Muscovy fighting over each other.
The "Tartar yoke" had a major influence on Russian re-unification, but not so major as to cause it. Really, it was a logical process: in the middle you had the semi-feudal principalities that could and would grow. On the northeastern peiphery, the merchant republics were not expansionistic and would eventually be gobbled up by the power rising in the centre. Add to that a common language, culture, and religion and the gradual unification was pretty damn likely. It doesn't have to mean an effective central government: during the middle ages, most states had at least periods of HRE-level fragmentation (and the HRE had periods of greater effectiveness): it was simply Germany's bad luck that it remained that way owing to a variety of factors, and of course one can argue that Russia was still that way until at least the Time of Troubles.
In fact, you might actually get a more "unified Russia", sooner: you'd have to ask an expert on that side of the thing, but without the Mongol invasions would so much of White and Little Russia been conquered by the Lithuanians? Furthermore, Moscow is less likely to emerge as the centre of gravity. It might saty with Kiev, which lost much power and influence because of the Mongol devastation.