Phewwwww.
Thank you. Nice to get good news after about 36 hours of pure panic, complete with literally staying up all night trying to sort through paperwork and ideas.
The ESTA is just the electronic equivalent of the green I-94W (visa waiver) landing card you used to fill out on the plane.
A few years ago, they stopped issuing landing cards and moved it all online, so that you got approved before you got on the plane. So now, instead of filling out the green i-94W card each time you fly, you just submit the ESTA online once every 2 years instead. You only need to redo the ESTA before the 2 years are up if your circumstances have changed.
Question:
If my fiancee has the ESTA, is there any possibility she'll encounter trouble with an immigration officer on the way into the US? Normally I would say probably not, but we're now trying to plan ahead for all eventualities.
She shouldn't do, but she'll probably get the normal questions: Purpose and length of visit, where she's staying etc.
Would it help our case for her to book a hotel for her stay, or is it enough to give my address and phone number?
She just needs AN address where she will be staying - it can either be your address or a hotel.
Are we breaking any of the conditions of an ESTA to get married (in the US) while she is visiting on it? From what I understand, it can't be "planned," so I assume things like booking a premarital course (to do away with the 3 day waiting period after applying for marriage license) is out?
Nope, the US Embassy specifically states that if she wants to marry in the US, and not live there afterwards, she should do it on the ESTA.
See:
https://uk.usembassy.gov/visas/fiancee-2/From that page:
Please note: if you do not intend taking up indefinite residence but will continue to live and work outside the United States after the marriage ceremony, you should apply for a B-2 visa, or if eligible, travel visa free under the Visa Waiver Program.