How much snow do you get where you live? Small folding bike wheels could be a pain coming out of ruts and drifts. An old rigid frame mountain bike would probably suit your needs quite adequately. If you don't have any steep hills, consider using a single freewheel or an internally geared hub to avoid a derailleur becoming frozen in one position with ice buildup. I've ridden my Alfine geared bike in temperatures down to -28ºC with no shifting problems, even through drifting snow.
E.T.A.: cleaning your bike regularly should prevent any corrosion problems. Probably best not to do it with a high-pressure spray like these guys, though.