This question is to help me find peace.
First, the question of the Snowflake's compactness has been tackled here on this site:
But I'm inclined to believe it is not. The only reason Henning Makholm's argument does not fully convince me is because I believe I can come up with an open cover without finite subcover. What you do is surround each triangle with an open triangle small enough so that it doesn't intersect the 'sprouts' two levels down which sprouted out of the first iteration, and then repeat that ad infinitum. I hope this picture helps (the grey lines represents the open sets):
So, my question is why does this open cover have a finite subcover?