The seat next to the driver is called the "front passenger seat". Passengers at the back of a car are seated on "back seats" or "rear seats". In other words, "back/rear seats" are found behind the "front passenger seat" and the "driver seat". Just as a side comment: Some cars have back/rear seats that can be folded to create a flatbed configuration. These are "fold-down back/rear seats".

Next to the driver: generally speaking, "front passenger's seat". In conversation often you will just say "front seat" or "front" and from the context they will know you mean next to the driver and not the driver's seat. For instance, I might say: "Let's go. You can hop in the front with me." Seat in the back: "back seat" or "back" for short. As an example, I might say: " You two can hop in the back with Mary in the front." So two use the back seats and Mary uses the seat next to the driver.

Yes. That is correct.

In American movies, riding in the seat next to the driver appears to be called "riding shotgun", or "shotgun" for short. Probably related to when coaches needed protection as well as driving. . This position is also often a navigator. So ocassionally referred to as that role. I'll navigate" - less common.