You messed up when you gave a number. It's rare that you absolutely
must give a number to proceed with the interview process, so it's best not to. The companies almost certainly did not make offers that can't move up, but you've capped yourself already and if you try to ask for more now you won't look like a competent professional.
Agree that you can try to tell them you have other offers but there are other ways to squeeze more from them too. Make a list of the benefits that matter the most to you (signing bonus, vacation time, work-from-home days, continuing education, etc.) and ask for them in order of importance. Chances are they have wiggle room. If they say they can't budge on X, then you can ask "Well, how about Y?".
As for what's on a background check I think it depends on where you live and the company doing the background check. Last time I went through one they called my last two employers and confirmed when I worked there and how much I made. I believe it was
this company.