Now I'll touch on some common application and/or interview questions that may be burdensome on transgender applicants (applicable to the U.S. - and since I'm not a lawyer or HR worker don't take what I say as legal advice; it is from my own research and past discussions about this on Susan's and elsewhere).
The first one is can an employer ask the prospective employee to state their sex? The only time that this would not be a "taboo" question is:
1. Sex (not what gender they're presenting as) is a bona-fide occupational qualification. This would usually be limited to athletics, jobs where you or others around you may be nude (e.g. locker-room attendant), etc.
OR
2. The information is asked on an optional survey (which would typically be accompanied by questions about your race, ethnicity, etc.). If you don't want to draw attention, just opt out of the entire survey.
If asked any other time, follow the advice given by me and others here.
The second one is can an employer ask for former/other names the prospective employee has used? The answer is only if it is necessary to properly verify information provided by the applicant. Since most employers these days will run some kind of background check, that unfortunately means that for most transgender people who transition much past coming of age (and thus would probably have something under their former name that would be relevant) the answer is yes they can ask for that information.
If the employer needs to know your former name to verify your work history, educational credentials, or contact references, you risk coming back with a false negative (that you never worked or attended school there) if they can't verify you because your record is not under your current name. You might be okay if it's something they're unlikely to verify (such as high school once you've graduated college or employment from more than a decade or so ago).
The bigger risk is with something where not having all of the names the history is under would indicate a false positive, such as with criminal or credit history checks. If you end up concealing anything along those lines deemed relevant (even minor offenses for example) because you didn't give the name the information was under you could be accused of deception/lying with the consequences associated with such.
That means that with the exception of teenage/earlier transitioners, or in some cases those who transitioned many years ago, there is probably no way to avoid outing yourself via your former name to a prospective employer (unless you can send the sensitive information directly to the agency or investigator; but if you can't do that, or an employer wants for example to contact a former employer or school directly, you basically have no choice unless you can get the institution to update your records with your current name).
For those who transitioned either before (or not long after) becoming an adult, you may be able to put the "alternative response" tactic to use - if you can honestly say that you have nothing relevant under any other names that answers the question as far as they need to know. (In cases where the applicant was adopted or the name was otherwise changed as a child they generally say that you don't need to even count your birth name, but in the case of transgender people where an employer may find any excuse putting a response like I suggested CYA without outing yourself.)
Someone will likely point out that there may be exceptions, such as a security clearance, where they want everything from birth (but there from what I've been told by another member who I don't remember her name on here the clearance investigation is handled confidentially by an agency separate from who actually makes the hiring decision - and you can put down that the sensitive information is available only to an authorized investigator). Another exception may be in certain cases, like the kind of work our Jessica Merriman works in as a paramedic, where because of the nature of the job a deeper investigation than what can normally be done - she once said that applies to any kind of state/local government job, but in most cases the same rules that apply to private employers also apply to state and lower government agencies (it's her specific job that makes it different for her - someone like a police officer would likewise fall under heightened scrutiny, but run-of-the-mill jobs like a park manager or maintenance worker wouldn't). The federal government is exempt, but there the OPM has a way to handle the checks without outing yourself to the one making the hiring decision (if not otherwise relevant for reference checks, etc. that are handled directly by the agency employing you).