SPCR

what kind of adhesive to use to adhere sound damping materials?

the cheaper the better but it has to work. is there anything out there that is strong but yet easy to remove in case of a mistake or future change/update to a case design?

Most sound-damping materials already have a pressure-sensitive adhesive on them, no need to buy a separate one.

The Peel & Seal, will not likely remove very cleanly and would require a solvent to remove residue. I can't speak to the higher end stuff, but they would likely use non-removeable waterbased PSAs.

The idea with this stuff is that you don't want it to come off, especially at the elevated temps inside a PC case.