PhD application: my country isn't listed
The United States does not recognize the statehood of Palestine (meaning technically that can't be your home country as far as that government is concerned), which makes it somewhat unsurprising that an American university might not have it listed as a country.

The Department and School likely doesn't have a particular stance on the issue - they probably didn't even design the application. Of course the ideal option would be to call them, but as this isn't an option at the moment, I'd put something close (Jordan - which as you mention you hold a passport for) and then call them to clarify as soon as you are able to do so.

Also, as this is clearly a recurring problem, you should seriously consider finding out if a given school's application has this problem well before it is due.