as far as background checks go the answer is it depends. i changed my name in march and this is all from what i understand. I live in and was born in texas so thats where my stuff was done. most background checks will consist of just a criminal check and as long as you have never been arrested or have a record theres a chance they wont find out. however, if your job includes driving , they might run a check on your drivers license. your name change will pop up on this becuase your license number doesnt change, at least here in texas.
as far as your social security card, if you have had a name change i doubt it will come up because your name would match. i changed my gender on everything including my license and my social security card. regardless of what laws apply, honestly it all depends on if there is a ts friendly judge in your area that will put out a court order of gender change ( and im pre op and got this done ). go find yourself a good lawyer and get the name and gender done at the same time. my advice would be to get ahold of a ts group near you and ask around about lawyers that will help with gender marker/name change stuff. my lawyer cost 600 dollars and it was all extremely painless. 🙂