It’s an easy task, travel from Damam to Doha, but u have to follow the road signboard of Saudi which are very good, if you’ll drive the car then I think car should be in your name, no need to bring any thing from Dammam to Doha every thing available in Doha. From Qatar border to Doha city only one road comes (Salwa Road) u can’t go here and there. If you have idea of Dammam,Hofuf then very good.
u must need visit visa of State of Qatar.

Good luck and have a nice journey.