Until you've signed an offer letter you don't have the job. If it works out great but until then keep applying other places.
This happened to me. A headhunter reaches out to me on a Saturday morning. He put together a phone interview that afternoon. Brought me in on Monday for face to face interviews. Landed the job. I used the headhunter as leverage in working on getting a signing bonus, stocks, and placed into a higher bracket for year end bonuses. All came out to a total around $20k extra. Those stocks are now $60k.
Also, if you require to move ask for relocation funds.
Anyways good luck hopefully you get a nice package to sign.
So yeah, expect something to "come up". Shit always happens.
There’s always something sketchy about places that reply insanely fast after applying..experience shows that places this desperate for quick hires usually have insane turn around or they just need a body to do crap work. In any case good luck.