I think small LAC’s would be the most friendly to LGBTQ students. My oldest daughter looked at a few that really went out of their way (for example gender-neutral housing), but most LAC’s (non-religious) lean liberal. But since you are understandably wanting a bigger dating pool, you could go with a big U or a consortium or a city with several colleges. I would say most U’s lean liberal too, but not quite as much (at least judging from oldest daughter’s college search). Tech schools tend to be more conservative, so even though you are sciency, you may want to consider that.
I would suggest looking on college’s websites to see if they have any LGBTQ clubs or such. Personally, I’d also suggest avoiding “red states” unless the college is in a more LGBTQ-friendly city (say Atlanta or Austin).
As far as private schools on a budget – once you get your scores, look at the mid-50% scores for the schools you’re interested in and concentrate on those where you are above those stats (or at least near the top). That way you are more likely to get scholarship offers.
U of Idaho is large enough that it might be OK. Moscow is a nice college town.
I’d take a serious look at Western Washington - it’s a WUE (Western Undergraduate Exchange) school and your stats are probably good enough to qualify for the WUE OOS discount. It has a very socially aware student body, has non-binary/gender queer housing available, and is not too far from Seattle. Large enough to find all types of students. Don’t know about their financial aid for OOS students but with WUE tuition, you’d pay about 20K for room/board/tuition which is just slightly more than in state for U of Idaho. Maybe less if you qualify for financial aid.
Sit down with your folks, run some EFC calculators for target schools, and tell them what it’s likely to cost. If they could scrape up 10-15K between them, you could finance the rest with a loan and part time/summer employment.
Room, board, and books is likely to cost about 10K+ wherever you end up. So if your stats qualify you for full tuition, you or your folks would still have to come up with living expenses.
Best to have this conversation with them now so they can begin planning realistically.
Good luck!
p.s. My D17 goes to a high school with about 80 kids per grade and the gay students don’t seem to have any trouble finding dates. At a large U, you should do fine.