With no or less Mongol invasions, a thriving Ukraine might lead to more European cultures in its area?
Term "Ukraine" in its modern meaning does not make any sense in the XIII century and there is no guarantee that it would end up as a meaningful entity in a changed history.
There were:
1. Great Princedom of Kiev - by the late XII, while still formally the 2nd most "senior" princedom of the "Rurikid Empire", losing most of its importance due to the proximity of the Steppe and raiding nomades (Pechenegs, then Polovtsy to be incorporated into the Golden Horde).
2. Princedom of Chernigov and Princedom of Pereyaslavl - both parts of the "Rurikid Empire" and suffering from the same neighborhood.
3. Great Princedom or Kingdom of Halych-Volynia (some of the rulers adopted royal title) - while culturally predominantly "Russian" (in the general terms of language, religion, etc.), in close contact with Hungary, Poland and Lithuania. IIRC (but don't quote me on that
), formally not a part of Kievan Rus.
It just happened that eventually all these territories had been annexed by the expanding Lithuania (together with what now is Belorussia) with some of them ending in Poland and forming, for a while, something of a single entity.
"Ukraine" means "border lands" and the terms had been for quite a while used to describe not only parts of the modern Ukrainwe but also the territories of the Muscovits state bordering with the steppe. Now, about the prosperity and "European cultures" part. "European" (as in "not nomadic") border remained practically the same from pre-Mongoliam period and all the way to the mid-XVIII when the Crimean Khanate was annexed by Russian Empire and the raids ended. In other words, the Mongols/Tatars were not a critical element as long as there were some nomads on the South. Areas removed from the steppe (in the modern terms "Western Ukraine") had been in a better situation than the border lands even if the border lands are presumably more fertile.
In OTL Polish colonization of the border regions resulted in a series of the popular unrests and eventual loss of the "Left Bank" Ukraine and creation Hetmanate state as a vassal of Moscow (actually, it was more complicated but this is irrelevant to the issue). Western Ukraine was much more "polonized" and one can see clear differences in surviving architecture of the late XIX century: it looks rather typically "Austrian", which is not surprisingly. OTOH, the areas of the Eastern Ukraine (including the former nomadic areas) are more Russian (in case you missed it, Russia is also "European"
), which also not a big surprise.