Interesting questions, both of them. The second one is easier I think: should the runner ask, the corp has to reveal all face down cards at the end of the game. Since both players play corp, both reveal similar info, so no unfair advantage next time they meet. This protects against cheating and misplays (which is more likely I think).
As for the first issue you raise, I think hard rules would be tough to implement. As long as the corp makes sure they spell out exactly what they are doing, and like you said don’t shuffle remotes around, I think the onus lies on the runner to remember which serveris what. I think the “no note taking” clause of the rules kind of covers this.