Thin tires may not be as much of a problem as you think - I've been riding through the winter for several years on a 90's mountain style bike using pretty thin tires - 26 x 1.50 - and they have much better traction than you might expect in snow and on ice. I do ride defensively if the roads aren't clear though - a little slower, and if there's a lot of snow I do try to avoid heavy traffic roads. That's more due to lack of confidence in the automobile traffic, plus splashing slush if its warmer, than worry about crashing.

Regarding the single speed - if you want to try it out without any cash investment you might be able to lock your derailleur into one gear with the adjusting screws - if you don't like it you can change it back. Fenders? I have a set of Zefal or similar thin plastic fenders set up with plenty of clearance - they work well and I haven't had any trouble with snow/ice clogging, as I recall they were less than $20US and pretty easy to install.