In researching your question, I learned, or perhaps unlearned something. I had thought there were New Testament descriptions of the birth that mentioned baby animals like sheep or oxen or an ass however in reading today I learned there is no such mention, only that there was a manger which is an animal food trough. This relates to your question as I had been told long ago that the presence of baby animals was the proof that Jesus had to have been born in the spring as that would be when baby animals were around. I also learned today that mangers often were found on the first floor of homes while the usual sleeping quarters were upstairs so maybe no stable and no baby animals
Was Jesus born in a stable?
and that the birth story as most clearly told in Luke about the family having to travel to Bethlehem because of a census being taken by Quirinius while Herod was king is historically nearly impossible. Perhaps more translation problems
Census of Quirinius - Wikipedia, the free encyclopedia
Thanks for posting the question which lead me on this little educational journey. Didn't find a clear answer to your snow question but I think those above seem very reasonable