>How much contact did the population of Finland have with Mongols during their age of conquest?
next to none
The meme i assume comes from the language, Finnish is not related to the rest of the scandinavian languages, in fact, is not even indo-european, is from the finno-ugric language family, Hungarian and some siberian languages are also part of the same family, so i guess it comes from there.
Also, most populations in the north and i mean the farth north, are composed of "mongoloid" people so finns seem to have some admixture from those populations.
i'm no anthropologist, so take that with a grain of salt