Others will speak in general, I’ll try to respond specifically for Middlebury.
For colleges like Midd which have more holistic admission, they could help. For UCSD, not much.
For most high schools, that would make it almost impossible. But from Stuyvesant you may still be in the running, though it would still be a challenge.
Asians are underrepresented at Middlebury, so that won’t count against you.
Unless a teacher is going to write you a LoR which is at leas a 9/10, there s no reason to ask them. Colleges with very low acceptance rates look very closely at LoRs. So if you cannot get at least two teachers to say that you are an amazing student who would be a huge addition to the college, that does not make for a very convincing application, especially if your GPA is on the low side.
I’ve only taught college students, and I’ve written LoRs for students applying to grad school. I have written LoRs to two students who got Bs in my class. The only reason that I wrote those letters was because I could truthfully write that the grades did not accurately reflect these students’ mastery of the material, and that I could confidently say that they would be wonderful additions to their respective programs (both were accepted).
A 5-7/10 LoR will not help you get into a college with low acceptance rates, and you should look for somebody who will write you a 8-10/10 instead.
You should also be asking for those LoRs now.
Comments on other colleges:
Neither UCI, W&M (OOS), nor Bates are even close to be a match for anybody.
If your parents are not wealthy, they cannot afford any UC as an OOS student.
WOuld you be happy to attend Bing if that were your only acceptance? If not, it isn’t a safety.
It also depends on whether these colleges know Stuyvesant. People on the East Coast know that it is one of the top public magnet schools, but in the Midwest, they may not be as familiar with it as they are, say, with Whitney Young or IMSA. So that 3.7 GPA could make admission more difficult.