I have built and lived in small cabins like this, and even an old army tent, for years in Georgia. A few hints:
----as mentioned above, a big woodstove in a small space can make up for any number of deficiencies, provided you've got plenty of wood to burn.
---yes, layer the roof. At the least, a second layer of waterproofing. I've used old
mattress bags out of dumpsters and other large pieces of plastic, old tarps, etc. laid overlapping so as to shed
water from any holes, etc. downhill. The heavy, rubberized fabric tarps used by truckers and large tent rental companies are the best, but hard to find. Then put something else up there to keep the sun off the tarps. Old carpets work great for this. You can tie these to each other by using a heavy "needle" made of stout wire and baler twine or some such, and if you really want to get crazy then stucco it with cement. That will stiffen in the carpet and make a solid, storm proof roof. Carpet is a poor homesteader's equivalent of buffalo hide....so versatile. The other cabin was made with "panels" of studs with a hollow space, and carpet inside and out, and the space filled with whatever for insulation.....whatever the dumpsters would provide....egg cartons, styrofoam, bubble wrap, carpet padding. This stuff won't mold as readily as organic stuff, but I guess you could use that too. I would think about adding some studs or more poles to the inside and then an inner paneling of something, even
cardboard, and stuffing the space with insulation. And get something on the floor....rugs, sacks, etc....it will make the space feel a lot warmer. If it's an earth floor, you will gain even more heat by making a tunnel under the floor from the outside so as to lead outside air directly to the intake of your woodstove, using some thin metal ducting or some such to bring it up to the openings. That way your stove isn't sucking it's own warmed air from your space to
feed itself....