Chester and Sarah, I notice no one answered your specific questions. I won't either.
It may be too late as I suspect you have ordered the stove, so I may not talk you out of it, but there's a lot of work involved. I've been supplementing heat with a wood stove for many years. First of all, it's a mess. Carbon gets into the room from the burning wood, a little at a time, but it does, and makes things dirty. The wood is messy. If you cut it yourself, you'll figure out how small to make logs so they fit in the stove after a while. Maintaining a fire means someone has to tend to it every few hours—or less—depending on hard or soft wood. Hardwood is more expensive if you buy it, but it's less tending to the fire. Felling trees and cutting them up is hard work and chain saws require lots of maintenance. Hand saws work fine, but time is a factor. I still do it, but not so often any more; you have the advantage of being 45 or 50 years younger. Maybe you've done all this; if you haven't, you may not like it.
No water tanks, but water jugs I suppose. They will freeze as well as the toothpaste and other things you never thought of. It takes a long time and a lot of BTU's to defrost a plastic jug of water—can't get it too close to the stove. Maybe not every day, but you'll possibly feel tethered to the trailer, and not wanting to go away for a few days. The black water tank will freeze or whatever you use for a toilet and will be no fun to defrost (if you use the Airstream toilet, there will be after a while of column of frozen "stuff" reaching ever closer to the flap in the toilet itself; it will not smell good on warm days). I've know plenty of people who wintered in cabins with wood heat and no running water and it was a high maintenance lifestyle. They acted tethered to prevent everything from freezing. I think you will find you are combining the propane heater and wood stove to provide some heat around the clock. Newer electronic thermostats can set the furnace to just heat the trailer to 40˚ and don't use as much propane and then you can supplement with wood heat. Look into a free standing larger propane tank—120 gallons maybe—that can be delivered and you won't have to take the trailer tanks to get them refilled all the time.
Without a fan, the heat will be concentrated around the stove. At the far ends of the trailer, prepare for really cold temps. In the winter, the PV may contribute little—there can be days of dark clouds in Vermont. Don't park under trees and obstruct the sun.
The trailer will not hold heat very well—it's got some insulation, but it sure has a lot less than most houses. And some models have a lot of windows—all single pane. They get cold very fast. A catalytic heater is more efficient than the furnace and some people use them for fulltiming.
It's a romantic idea and I don't think you are crazy, but it can be stressful. I applaud you for trying different things for an exciting life requires experiments. You'll learn a lot about yourselves too. Seeing you are college students, if you are serious about your studies, time is a factor.
You're lucky to live in Vermont. One of the more beautiful states (especially if you like clouds).