trans | Disclosure and the hiring process

Gah, okay, so I recently started a new job (I just finished Day Two). I've legally changed my name and sex, and so far there's been no mention of my gender/trans status, so as far as I know I'm "stealth" or whatever.
However, I haven't handed in all the forms yet, including the bit where it asks if I've ever worked under any other names. Now, neither this company nor my city/state has any anti-discrimination protections for trans people, and fwiw the owners of the company are Orthodox Jews *shrug*

So I mean, I'm already settled in there and they seem to like me, and it would hopefully be fine, but... I guess what I'm asking is, I basically have to come out to them, right? Lying about it and hoping they don't research it would be a bad idea, right? (But also fwiw, it's total entry-level data entry work, and all my previous work experience is stupid summer jobs like delivering pizza and crap, nothing they'd be likely to care about checking up on...)

And, assuming I do end up putting my old name down, I should probably talk to them about it, right, and not just force them to ask me when they get confused?

I'm just really stressed, and this is my first Real Job ever, and I don't want to mess it up (either by getting fired for being trans or for trying to hide being trans). What would you do in my position?