1. Most of us have gotten away from mineral spirits and replaced with "friendlier" and less flammable cleaning products. Heck, mineral spirits aren't even that much cheaper any more.
Petroleum based cleaners have also been known to not play well with some emulsions especially with those underexposed, iften adding to a difficulty in reclaim. Moose already pointed out the need to test how different solvents react with your emulsion(s).
Back in the day, Xylene was our common cleaner, and the new photopolymer emulsion from Murakami didn't react well in our shop. We kept and reused a lot of screens and hallftone areas required caeful inspection. (screen opener was even worse)
2. Have you tried tapes with less residue? Clear tapes with rubber based adhesive are popular and easily available. They still require some cleaning, but tend to not leave a stcky reminder. Heck, you may even get away with only pulling the reg mark tapes on one screen with clear tape over the rest.