I live in a travel trailer in Idaho and it's okay except for insulation. One space heater heats the whole trailer (it is a 29 foot trailer), though. A heater coil for the outside water hose is a MUST if you don't want burst hoses and pipes and no water. Good insulated skirting helps a lot.
I have propane for the furnace, stove and oven, and a refrigerator that runs off either propane or electricity. The overhead lights are 12 volt and, when the power goes out, the batteries take over for the lights. I could add more outlets to run off the battery and an inverter so I could plug something in if power is lost.
If I could afford it, I'd build a little room off one of the doors and put in a small wood-burning stove. We have national forest all around us, so wood heat could be free, or nearly so, and plentiful. (For a $10 fee, you can get 5 cords of wood free each year.) I'd also get a solar panel.
Since the trailer is metal, a dehumidifier is necessary because of condensation that builds up when the outside and interior temperatures are too different. It actually can drip from the windows, doors, and skylights.
Another thing about a trailer is the lack of storage space, although the whole underside of the trailer is available with skirting in place. I'd like a little storage shed added to it.
Travel trailers are built to be self-contained for at least a short time, with a cold water tank besides the hot water tank and a water pump (which uses electricity from the batteries to operate), two propane tanks (EACH of which lasts me at least a month, although I operate off a 100 gallon tank now and don't use the small ones), a switch to turn off the propane hot water heater until you need hot water, and the other propane utilities. The batteries are two deep cycle batteries that are constantly recharged when there is electricity. I don't know how long they would last if there is no electricity to recharge them. So, in that respect, they are better in a SHTF emergency than most houses. I could go several weeks without electricity, and I could take it somewhere else if necessary.
You can put a false roof over the trailer or, as was suggested, park it under a canopy or inside a barn or shed.
Something about northern climates that most don't know if you don't live there is that there is a wide variation in temperatures within a small area, usually depending on altitude. I live in a valley which is hotter in the summer but warmer in the winter. The outside temperature almost never gets below 10 degrees F, there is very little snow, and most of the real "winter" is in January and February. My sister, however, who lives 30 miles into the mountains has very long and severe winters and a very short growing season. It takes months for the accumulated snow to melt, and you get around on snowshoes. Doors open inwards too because the snow may get too high to open them otherwise.
Two things that become important living in the cold climates are what is called the snow line and also whether you have a north or south facing home. The snow line is different in different areas: on one side of the line if it is rain, it falls as snow above the line, which may be only an inch away. I would want a home below the snow line at least. And a south facing home and driveway. Others may opt for a home higher in the mountains where it is more inaccessible to passing strangers. You may need a snowmobile to get to it at times, but always at least a four wheel drive vehicle on many roads. They don't always plow those roads.
All in all, I'd recommend a travel trailer, but be aware of the shortcomings of them too.