-
You could use fusible interfacing, which would basically glue a very thin layer of stiffer fabric to the T-shirt fabric. That would make the T-shirt fabric easier to work with, and more fit for purpose as a cap fabric.
I haven't done this before with old T-shirts, so I can't guarantee the finish, but it should work, theoretically.
If you're very keen to give it a go, let's chat in PMs to arrange things?
The short answer is no, unfortunately.
I'd love to make upcycled casquettes, mind you, and for exactly the reason you touch on - there's a lot of nostalgia inherent in an item you've had for years.
But T-shirt fabric is an absolute PITA to sew with, and it's not an ideal fabric for making caps/hats because of the stretchiness.
Any piece of meaningful clothing made of something close to cotton twill (like chinos) or cotton poplin (like a button-up shirt) and I'm all in.