You shouldn't see the gender question on a job application unless it's on a voluntary EEO survey form (or the form is asking something that it shouldn't*). What I'd do in the case of an EEO survey is since it's voluntary is don't answer any of the questions on it (don't answer everything but the gender as that may raise more eyebrows).
As for your name, unless you're getting a job like those that illegal aliens would take, you will at some point have to give out your legal name and SSN to an employer. Otherwise they will run into problems when they report your income to the IRS and SSA. In addition, even once you legally change your name, if you have any work, school, or criminal records under your former name that they want to check you may have to inform them of that (or else there will be snags when they run such checks). While in many places (but surprisingly not California) an employer asking for any former or other (e.g. religious or pen) names without a reason is legally dicey*, it is unequivocally legal for them to ask if any records relevant to a background check (i.e. work, school, credit, or criminal) are under another name. That means that in the U.S. the only TSs who may be able to work around not giving out their former name if asked are those who transitioned before having any "official" (where they send you a W-2) jobs or other "adult" records (the same would also apply to those who were adopted as children and had their name changed for example), or those who manage to get ALL of the aforementioned records changed over (if either of those apply I recommend answering the question "none that any relevant records are under" for cover).
A good analogy I thought of recently with the former names question is the "ban" on Americans traveling to Cuba: The courts ruled that the government cannot outright ban American citizens from traveling to a certain foreign country, but they can ban the use of American money there (meaning that unless you have income from a foreign source and can prove that you're only using such money, it's still a de facto travel ban). Likewise with TSs and their previous name, while there are restrictions on asking for any previous or other names without a reason, they can ask for any names that records they can check are under (once again except for the particular groups mentioned in the previous paragraph it still means that for all practical purposes you'd need to inform them of those names).
*Exceptions are federal jobs and certain other high-security jobs (the federal government is exempt from the usual suite of "illegal" questions).
