Foam & Cork seem to get the best reviews for insulation. If you use multiple sheets of thin stuff, it is more flexible and contours to curves better than the more rigid thicker sheets. I’d go with whichever is lighter weight.
Showers can range from extremely simple to overly complicated. For mine, I use a hula hoop for a shower curtain rod, with regular shower curtain hooks & curtains, all from the dollar store. This can be hung from anywhere, either inside or outside. Mine is also my toilet curtain. Then I use a storage container to catch the shower water. I use a weed sprayer with a kitchen sink sprayer for the shower head, which requires no power and easily used either inside or outside. I even use it to wash my van. There are also Battery/USB/12v options available too.
Climate control with a pooch is a lot harder, especially for heat. I use a kerosene heater myself, but it is not pooch friendly to me. You need heat which is pooch proof, and can’t set him on fire. For this reason, I’d suggest an RV type furnace.
Cooling is easier, besides shade & ventilation, you only have 2 choices, either A/C or a swamp cooler. A/C is going to require either shore power or a generator. I’ve had better luck with a 12v power sipping swamp cooler. (homemade) It’s cheaper and uses a whole lot less power, while being equally or possibly more efficient than a regualr A/C. The only downside is it they do use water, but mine can freeze you out no matter how hot it gets outside. I’m the guy wearing a sweater inside my van when it’s 120°f outside. Since I both live & work inside my van, comfort is a big priority for me.
Most people tend to put a big storage garage under their bed.
"Proper planning, preparation, and equipment, ensure proper performance." ~ Road Warrior