Looking up the climate nowadays for the Bohemia region of the Czech Republic, the temperature in January goes down to about -4C, and the snow depth is generally no more than 30-40cm. As Nigel says, the peasant lived beside the forest (it's extensively forested) and would have had oodles of firing for his hovel. I think the song was written in Victorian times (by a clergyman) and the words mountain and fountain, hence and fence were selected in order to rhyme. The peasant's name wouldn't have been recorded anyway (assuming he had one, as in the 10th Century, such folk were just called eg 'Karol of the Forest', or 'Bedrich the Woodman' and so on.)
Wenceslas was respected for his charitable works, but nothing specific has been recorded historically. His name was actually Vaclav.