US Immigration from Philippines

I understand that the stamp placed in their passports when entering the US, serves as green card until the real deal arrives. My question is: can they travel from PI to US, go through POE, spend the night, and then proceed to Mexico. I have a house in Cozumel, Mexico and due to lack of housing in the states until a later date, I wanted to bring them there for a couple of months.

On my family;s St Luke;s list is "Letter with Case Number", what exactly are they talking about. In their packets are the St Luke appointment letter and Embassy appointment letter. What other letter do they mean?

Are there certain POEs to avoid when coming from the Philippines. I was thinking maybe NYC and LAX should be avoided. Any experience out there?

Hello my fellow 751 sufferers....My wife has a 751 pending since 09-15-2021. She was ambivalent about applying for citizenship. She is finally taking the N-400 plunge. For those of you that submitted the N-400 online how did you let USCIS know that there is a 751 pending so it can be pulled and routed with the N-400? Thanks as always...B.

We won't be able to file the N-400 for about 6 months yet, and a good bit of that will be out of the country, but is there a domicile requirement for the N-400? And what would be acceptable as proof of that? We're full time RVers, so addresses are kind of difficult. My mail goes to a friend's house in Ohio so we have some financial docs that have that address but do they need more, like a lease?