It's always been runs scored. ERA would underestimate league-wide scoring, and errors =/= TRS events. The only way TRS incurs a double penalty (or bonus) is if you own a pitcher and player on the same roster, and the +/- TRS event happens while that pitcher is on the mound.
The answer to most of your MLB questions is "MLB does it this way because it's always been that way," which I wouldn't consider very analytic reasoning.
Regarding the ERA, Mendoza doesn't use ERA. It uses RA/9. It will always be >=ERA. And, yes, this is how it should be determined, because otherwise you're counting more offensive than defensive runs. The league-wide TRS should net to ~0, and is just an adjustment to RA/9 to account for your team's defensive quality compared to the defensive quality your pitchers received in their MLB games (the underlying assumption is your pitching staff on average plays behind a neutral defense on their MLB teams).
At this point though I'll just let Arne or others take it.