For the cotton that is placed at the bottom of the percolator...is it ok to use part of a cotton t-shirt for that? I have some old t-shirts that I was going to use for it, specifically the part where you place the cloth on top of the plant material.

Would it be ok to use part of a t-shirt for the bottom cotton part that the water seeps through?

Yes, cotton from a t-shirt (or similar) will work just fine for THP. The only thing that I would add is that with THP, flow rate is rather important. That is, you want the water to be in the percolator for long enough that it's able to extract the material before cycling through.

I tend to wrap my material in strips of fabric from an old t-shirt, or put it inside of an old (clean!) sock and then use either the end of the sock or an additional strip of fabric to ball up into the neck of the bottle so that the flow of water is restricted.