I use 6 mil Clarke.
I use white because in the Midwest vs. say the Pacific Northwest, it's sunny here nearly all of winter and you'll cook plants in smaller shelters like what you're going to build. You still will need be diligent about opening that up during warm stretches come late winter toward spring. On a very cold January day with full sun you can expect temperatures in there to be 70-degrees.
I've never messed with trying to keep a solitary tree warm with say Christmas lights. That's what most people use. If you don't have power there, one thing you can use are painted black drums filled with water that heat up during the day and give off heat thru the night. The other thing I like a lot is reflective insulation. Put that on the ground. Trying to run a small heater with a thermostat will work but it really racks up electrical bills.
My hoophouse is non-heated and the plants are up off the ground. I'll lose occasional plants but for the most part it works great. My greenhouse I set the heater to kick on at 34 degrees all thru winter. Even it being 16' long and 6' wide, I see winter temps at or above 70-degrees on sunny days. That's when it's 0 degrees. It's pretty well the same for 20 degrees and on the coldest days of winter at -10 I can still expect a sunny day to bring the temperature to 40-45 degrees. It faces south, is an attached greenhouse, plus the entire top of it is fully shaded because it's directly under my deck.
Some things for you to think about there. I agree about the 2 x 2's. Versus ripping you'll probably only add 10 cents to each board or a total of 20 cents, for example. That's a good idea. Your perimeter at the base should at least be a 2 x 4 though.