I'm currently an undergrad at Yale and I just stumbled onto this thread to see what people were saying. After living in New Haven for 3 years, I must say that it's no more or less safe than any other urban environment. There are really nice areas (west haven, east rock etc...) and areas that you really should never go (down whalley avenue, the dixwell community, past howe st. etc.). The housing in downtown around the green can be really nice and affordability ranges. The apartments on or near Chapel st. all have great locations and are generally pretty nice with prices that range from $700-$1100 per person in arrangements ranging from studios to three bedrooms. The Taft, Townsend, and the Warner are all pretty nice buildings. Lots of grad students live in downtown, in the grad school ghetto, or in east rock. I'm living in and around the lynwood st. area, which is also pretty safe, but I'm pretty sure only undergrads live there. HGS seems worse than undergrad housing at yale to me, although it has the benefit of being in the center of campus and it houses its own dining hall. This is a good resource: www.elmcampus.com
In terms of part-time jobs, Ithere are a bunch on listings on the student employment office website (www.yale.edu/seo) ranging from babysitting to library and part-time research jobs. I'm sure it's fairly easy to get a job at a local cafe or coffee shop as well and I've had friends who do that. Yale jobs, however, have a minimum wage of like $11 an hour which is probably higher than working at Koffee or something.
In terms of LGBT life, there is huge LGBT community at Yale both at the undergraduate and gradaute levels. Gay nightlife, however, is lacking although if you really want to go out, there's no reason not to go to New York. There is Gotham which has gay night on either fridays or saturdays and BAR is lgbt on Tuesdays. Other than that, I only know of Partners on crown and park st. and York St. Cafe which are both pretty sketchy. That being said, in terms of support, house parties, etc... yale is really gay-friendly.
Hope this helps!