If you were an immigrant with legal status who lives in the US, would you feel OK traveling? What precautions would you take, what information would you memorize?
Yes. I still think that those with student or work visas or green cards should travel internationally. It's just a question of being smart and carrying the right documents. The right documents will vary depending on the status so I can't give a list but I'd recommend having a brief consultation with an attorney. Many like myself won't charge for a 15-minute consultation, which is all that this should take.
I would guess unnecessary delays. Leases or property ownership can be verified with a phone call but probably needs to happen during business hours and if you are prepared everything will have a better chance to go smoothly. Property ownership is public record but a lease might be a bit more trouble to verify if the landlord is privacy conscious.