2" x 4" lumber is not 1 3/4" x 3 3/4", as the link you provide says, so you would need to readjust your math. That 1 1/2" thickness takes up space in the 16" spacing, so it is 14 1/2" between each stud.
As mentioned, some newer style construction uses 2" x 6" (1 1/2 x 5 1/2) studs 24" on center (22 1/2" between), but stupidly it isn’t as common as online articles would have you believe. Being in Florida with very hot temperatures and hurricanes, you would figure people would not make houses from under 1,200 sq. ft. to over 12,000 sq. ft. with 2" x 4" exterior walls because of the large energy savings and ability to use larger hurricane tie down hardware, but it’s pretty rare in my experience. The government mandates cars by efficiency, yet houses that use much more energy don’t have efficiency ratings. Construct as cheap as possible and pass the buck.
Another thing to note is the sheets of aerogel would have to be structurally capable of being used with an adhesive in order to get zero thermal bridging, as any nails or screws would be highly conductive. Maybe only a small percentage of the total area, but still a measurable number. It doesn’t look like it can be sandwiched without compression, and thus a drop in efficiency, but maybe other products have it sandwiched in structural panels.
Those links for aerogel available for purchase is great, thanks. If it had a slightly higher temperature threshold it would have even more uses. I wonder if it is the aerogel or the batting it is contained in that is the limitation.
Since you are looking at doing a tiny house, you might want to look into ones that have been Built With SIP’s. If they aren’t available locally, you could probably construct your own. That might not be feasible for a huge house, but with a tiny house you could make it exactly how you want it while laying on the ground and then just stand it up. I know I have seen a few on YouTube like that so a search should yield some results.
My understanding is that R13 insulation made for 4" walls is derated to R11 at 3 1/2". I think there are some fiberglass batts out there now which do slightly better. A big problem with fiberglass is installation. If it gets compressed and isn’t fluffed out again when installed it has a big drop in effectiveness. Having to cut and piece things together in some of the odd shaped bays can easily lead to gaps that allow thermal transfer from one side to the other. You can end up with an entire bay that effectively has about R1.
If this tiny house is fixed in place it might not be as bad, but one that gets dragged down the road will likely experience a lot of settling. The same goes for cellulose insulation. It fills better than fiberglass, but I would be afraid of it settling quite a bit if it is moved around regularly. This settling would lead to large gaps. I guess you could leave the tops of the bays open and stuff more cellulose fiber in as it settles. Spray foam and foam panels would not settle and would be a better choice for houses that see a lot of time on the road.
Unfortunately this ignores what most houses ignore, and that is the surroundings. If it is constantly changing then you may not have control over this, but the surroundings of a house have an Effect On The Microclimate which can lead to a significant loss of efficiency. I think there are calculators online which can show the energy savings of using trees to block the sun and reduce the need for cooling. Having shade to the South and West would be good in warm climates and bad in cold climates. Having a large masonry wall a short distance North could absorb sun all day and radiate that heat through the night. Keeping this in mind for a mobile house could help if you have choices of where to park.
Even something as simple as having a white or reflective roof in a warm climate or a black or charcoal gray roof in a cold climate can make a noticeable difference. I always see people so happy to get a great deal on cheap black roofing and wonder why their air conditioner is running all the time.
Permanent shutters that actually work and can be manipulated from the inside could also increase safety and efficiency. They could provide protection from road debris and prevent someone from breaking a window. When closed it could trap an air pocket to reduce heat transfer. Being able to open them from the inside would be mandatory in case of a fire or other emergency.
I’ve framed a lot of houses and been a fan of tiny houses for a long time. I personally like the idea of a small permanent house and large work spaces with plenty of space for gardens and trees and nature. If you have any other info about your build or take pictures along the way, then post them up! Good luck!