Re: Camper for deer camp??
The heating systems in most travel trailers run on propane. You will want to make sure you have a good, workable propane and carbon monoxide detector in the trailer. Then, you should have no problem in running one all night and keeping warm. However, if you do not have a generator or a land electric hookup, the propane furnace fan and ignition run on a 12v battery, and you can easily discharge 1 or 2 batteries during a long night (plus you can go through quite a bit of propane as well).
Also, you probably wouldn't want to carry water in the trailer's holding tank, use the toilet in the trailer or take showers in it in freezing weather, for fear of freezing the water lines and or holding tanks.
Also, keep in mind that towing a trailer of this size requires an adequately equipped tow vehicle, and winter towing with the potential of snow and ice can be challenging at times.
Assuming you account for these items, there is no reason not to consider a travel trailer for winter use. One should keep you warm and dry.
I do not know the answer to your question of camping on state lands.