Then anyone could have an affair just by rationalizing they were "teaching" someone
Well, what Remus is doing is pretty clearly teaching in a way that most sexual encounters aren't. Teddy specifically asked to be taught, Remus drew up an actual lesson plan and is assigning homework, etc. So he's not seeing it quite as a betrayal of Sam, but as something in a separate mental category.
As for diseases/protection - you're right, we don't really go into that (except for cleaning one's toys). Teddy mentioned that he used Construere with Victoire, though, so I would hazard that Remus probably made sure that Teddy knew about anti-pregnancy and anti-disease spells years ago.