Many countries claim successorship from Kievan Rus', most notably Russia and Ukraine. The formal origins of Russia or the establishment of a 'true' Russian culture seems to lie within various events between the 12th and 16th century. The defeat of the Khanate by Ivan the Terrible in the 1500s seems to hold the pivotal clues to the origin of Russia as a nation. Although the blurry history has led me to approach this question from a linguistic stance. From a linguistic perspective, the Ukrainian language contains heavy Polish and Russian influence. Subsequently this fact yields another question - did the modern day Ukrainian culture exist before the Russian culture or does Ukrainian culture stem from the original Russian culture?
The Russian word for Ukraine, Украина, is the formation of a preposition + noun, respectively У + Край (У - by, Край - border, edge). Is the 'edge (Край)' in relation to Europe proper or Russia? If the latter is presumed it would be supplementary to establishing the fact that Russia (i.e both the formal nation and culture) preceded Ukraine, in my opinion.
The history is interesting and complex. Unfortunately I couldn't find a definitive or satisfactory answer to this question.
Thank you
did the modern day Ukrainian culture exist before the Russian culture or does Ukrainian culture stem from the original Russian culture?
Ukrainian and Russian cultures both share the legacy of Kievan Rus'. Neither of them can really lay better claim to it than the other. Much like it doesn't make sense to talk about humans 'evolving from' chimpanzees, we can't really properly talk about Ukraine 'coming from' Russia or vice versa. In terms of language, we can talk about East Slavic-speakers from the tenth or eleventh century. Several centuries later, we can talk about Russian and Ruthenian, and a while longer after that it becomes meaningful to distinguish Ukrainian from Belarusian.
Is the 'edge (Край)' in relation to Europe proper or Russia? If the latter is presumed it would be supplementary to establishing the fact that Russia (i.e both the formal nation and culture) preceded Ukraine, in my opinion.
Originally it meant merely 'frontier', and East Slavs used it for a number of regions. As the Tsardom of Russia was getting on its feet, Polish and Russian speakers began to use it for contested land around the Dnepr/Dniepr/Dnipro inhabited by a variety of groups. That is, Ukraina was the border region between two fairly powerful states. As Poland waned and Imperial Russia waxed, the term Ukraina went out of style in Russia and was replaced by Malorossija 'Little Russia'. Early Ukrainian nationalists like Taras Shevchenko began using forms of Ukraine in the early 19th century, both in Russian Ukraine and in Galicia.
The first instance of Ukraine appearing on a map as any sort of organized state was in the aftermath of the First World War during the chaos in the area. Prior to this Ukrainians would generally have considered themselves Ruthenians, a word that also stems from the relation to Rus.
Your question about culture is an interesting one, because to answer it we have to recognize that despite the similarities that might have existed before the invasion of the Mongols, the historical events between then and when Russia came to dominate the Eastern European plains meant that the people of Moscow and the people of Lviv were exposed to very different events. Although it is not entirely comparable, consider the differences between Austrians, German Swiss, and Prussians in the 19th century. They are all 'German' and have very closely related languages (mutually intelligible in many respects) but culturally are very different. In the 19th and early 20th centuries the focus was on the similarities for both states, but since then the differences between those states has magnified. You can see a similar process happening in Ukraine.
So, coming back to your main question, did a seperate culture exist? Certainly. However, the differences between the culture of Muscovy that came to dominate Russia and the culture of Kiev and Lviv that came to dominate Ukraine stems from the historical exposure and separation that occurred, first under the Mongols, then under the various foreign powers that ruled over the Ruthenians/Ukrainians over the proceeding centuries.