1. You must ask your car dealer if itīs allowed to use your rental car in Czech.
I just know that itīs illegal to use some special brands in Czech when you rent your car by SIXT in Germany.

3. Not many refugees using a rental car to drive around europe.
Donīt worry about anything.
As long as you have legal papers with you what should happen?
Maybe they check your passport, thatīs all.