There is an assertion here that needs, first, to be dismissed, namely about unicorns: they were not part of folk belief. Unicorns appeared in medieval bestiaries and they appear in literature and art where they served allegorical purposes for the educated. The vast majority of people did not believe in them and they did not tell stories about them. If there was any level of belief, it was in a vague sort of way and only because they may have seen them in stained glass at the local church or some other encounter, but they did not tell stories about them and no one would have told a legend about someone encountering one. Unicorns were not part of European 'mythology' (I would not use that term; folklorists would call this folklore or oral tradition).

That's not at the heart of your question, but the example you put forward needs to be addressed.

In a post from long ago I took up the topic of fairies/elves crossing the Atlantic. This addresses in a generic way the question you specifically ask about the traditions and beliefs of the Scots Irish. There has little that has been done about that specific group of people. Scottish and Scots Irish beliefs are treated in Peter Narvaez, The Good People: New Fairylore Essays (1991), which includes an essay dealing with traditional Northern European fairy beliefs in Newfoundland.

There are some studies that address European beliefs in the supernatural surviving the emigration process. Ghosts, of course, did very well. In general, other supernatural beings didn't fare well, and yet the pockets of survival, when they did occur, are pleasant surprises to encounter.