Thanks for your research - I'm sure they are coached on how to answer all of these questions.
Regarding your above question. I was wondering the same. Again, I have heard, that they must ask for asylum in the first country entered, otherwise we don't have to take them.. So most of these would need to ask for asylum in Mexico. If they do not have to show proof, why wouldn't they just say they were Mexicans, thereby their "first country" would be the US & again, we would be obligated to take them in for processing & hearing?