Your desire to ride a track style bike in the winter is a little dangerous to me.... however-- a simple single speed-- front brake could be ok-- freewheel, good cruising gear... ok: however,
riding a fixed gear with track geometry is where its at, bro...... If you dont know what you are
describing-- a shop will sell you anything--- get in with some peeps who know this stuff on a regular basis--- take notes and study-- I would tell you if you are in a hurry-- dump that schwinn
and get a pista flip flop starter upper and be inspired for your next build-- learn geo. and ratios...esp vintage.....