How do I exclude admin strings from interface translation?
There is no solution that I'm aware of. You can't change the context of strings in core or contrib modules without changing the code. Which you shouldn't do.

The fact that toolbar doesn't show the english strings if your admin language is english is a big that's trying to be fixed here: https://www.drupal.org/node/2313309

I don't know if that's still doable in 8.x.