Ok friends, I have an update. They emailed (and SMSed!) me back asking for more details. The questions were legit (regarding my previous citizenship etc.) But I used the opportunity to ask them to cancel my order. Within 10 minutes got a replied back that the order was canceled (no questions asked) and indeed the reservation on my credit card is gone.

I can see similar stories here (i.e. when asked to cancel they cancel and give money back): https://www.trustpilot.com/review/www.usimmigrationsupport.c...

Just for me to sleep better tonight, I shall interpret this as a confirmation of the theory that this not a scam-scam, but just a business set up to trick idiots like me. And once they see that you realize - they just cancel in order to avoid confrontation. Cheaper and easier.