As for law schools differentiating which Beis Medrash is better than another, I doubt that they do that, mostly because it is hard to measure.
“Why? They could have a professor in the school who learned in yeshiva and knows the difference. Why would they not ask him to look at the transcripts and see if he can stereotype what kind of learner you are?”
If you have any reason to believe that law school admissions committees actually function that way, then that is good for applicants from Yeshivas. I don’t have any evidence of that kind of conduct though.