Yes, Oberlin is a great choice. I believe they have a transgender residence house, and stellar academics. Maybe try Earlham? I was really impressed by their approach to everything. They are apparently trying to grow their class size by about 50 per year (we were told), so you might have an opportunity as a transfer for some aid there.
Re employment, my sense is that things are getting better incrementally. Hopefully it will be even less of an issue by the time you graduate. I don’t think you are obligated to put the college you attended first year on your resume. But once you get past the interview stage, if a potential employer were to ask, you would be obligated to provide it. My hunch, though, is that many won’t ask. Some potential employers do make sophisticated social media searches though. It kind of depends upon what kind of positions you will be interviewing for.