In the US, Social Security will not change your gender marker without a surgeon's letter.
For MTF, it has to be bottom surgery, either orchiectomy or full SRS, with physician's letter stating "has completed sex reassignment surgery".
For FTM, I believe a carefully written surgeon's letter that doesn't specify the exact procedure but uses the right language "has completed sex reassignment surgery"will allow a gender marker change with top surgery or bottom surgery, either one.
The known by any other name depends on the employer. Definitely yes if there is a criminal or security background check required. For others, you could probably leave it blank, but saying 'No' would be lying and could cause you problems.