1. It complicates the immigration bit (a little) but not much. Nothing to do with the flying.
2. If you have a US PPL then you are entitled to a US radio licence. Get one at
the FCC online. They are not required for flying within the USA so most pilots never bother.
3. Yes, correct. You must also inform customs and immigration of your intended arrival so that they can inspect you.
And make sure that the aircraft you rent is insured to leave the country!
As for the UAE - I assume you are considering working there.......avgas is not freely available in many arab states and flying is quite tightly controlled.