You're not allowed to tear anything on the sabbath. But with toilet paper, that's more fanaticism. Since there are tears already, there are no letterings, and it's for hygiene, many rabbis allow it (I don't keep the sabbath after I started living on my own anyway, so I couldn't care either way).
Not touching the opposite sex, is to prevent you from fucking. That's why religious women have to wear shirts that go until their elbows, wear skirts that cover their knees, and married women have to cover part of their hair.