Cold weather won't negatively impact brewing, other than you may use a bit more propane. All the local brewers I know (in colorful Colorado) who use propane, including myself, brew in garages during the winter, to cut down on the cold, but it's really a comfort issue for the brewer more than anything else. I brew in my backyard most of the time, even in the winter, but if there's a lot of snow or wind, I move it to the garage so I can maintain a rolling boil.
Fermentation is a different issue. However, I ferment in my garage. I have a fridge (that was given to me) that's in my garage that has incandescent bulbs for heat and a temperature controller* so I can ferment at a specific temperature, then if I want to bulk age, I'll move ales to my basement or lagers to my beer fridge.
*I use the STC-1000. There are several different models, so you want to make sure you get the model with 2 controllers, one for heat and one for cooling. You can connect the power from the fridge to the one for cooling and either use light bulbs or a small heater inside the fridge connected to the heat power. Mine does Celsius, but I imagine that's what you'd want anyway. My fermentation setup cost me about $20, but I didn't pay for the fridge, and used a light fixture I had laying around from one I replaced. You can also get a Johnson controller if you don't want to mess with the DIY wiring, and don't mind spending a bit more money. UPDATE: There's a cheaper option if you don't want wire up an outlet box with an STC-1000, the inkbird. It's less than half price of the Johnson controller.
You could bulk age in the fridge as well, but so you don't hold up your pipeline, it would be best if you can move it to a closet (for ales) and you might be able to leave your lagers in the garage, maybe close to an inner wall as you don't want them to freeze, or maybe skip on lagering unless you get a second fermentation chamber you can leave at lager temps.