Unfortunately, I think it does, to a certain extent. It depends mostly on your potential employer and the state and local laws. Many employers do background checks on people. Legal name changes are public records so they can find out your old name even if they didn't ask. Otherwise, if you pass OK, you should be fine.
On the EEOC forms, I always check the female box - they are asking for my gender, not sex. Also, I say nothing at the interview about my medical history. It's none of their business which medical treatments I may have accessed in the past. I'm supported by HIPPA rules and New York State disability law in those areas.
When I get hired and the HR department begins setting up payroll data, however, I confidentially disclose to the HR manager about my medical history. Why? I haven't had GRS and the Social Security Administration (SSA) has me listed as a male, even though my driver's license shows female. I do this to head off any problems that could occur when the SSA sends a letter to the employer asking about the mismatch in gender markers. After I get surgery, my BC and SSA marker can be changed and that won't be an issue anymore.
I present my history to the HR manager and stress very strongly, that this is confidential medical information that is not to be shared with anyone without my permission. As long as you have been following the SOC you can legally do this.
Yeah, in most places they can still legally sack you because you're trans but the fact that you made it that far might just convince them to give you a chance. We need to get ENDA passed in the US House of Representatives so we can have legal recourse against being fired for that cause. Have you called, written and emailed your congressional representative? If not, do it now!