Congrats on the new to you car purchase. For the black interior if your wanting to maintain the clean, the product that he uses on interior is called Meguiars #40. its like wax in a sense its wipe on, wipe off. The longer you let it sit the more shiny it will stay.
As for the dull spots on the door, more then likely its just where the previous owner put their arm, so it just rubbed product off or got dirty. The #40 is a cleaner as well so if you scrub the area a bit with the product it can possibly clean the dirt out.
The touchless car washes are great in a sense that they dont touch the paint, so it wont scratch the paint. But in another sense its horrible for the paint because since hey are not "touching" the car, they have to use more harsh chemicals, which will strip every bit of wax or protect-ant off the car, leaving it vulnerable to the elements. Always prefer a good hand wash/dry using the 2 bucket method with a grit guard and PH balanced soap for the car.
But at LEAST the hand wash with ph balanced soap for the car to help the wax last as long as possible.
And when it comes to wax's, most over the counter wax's in Texas will last you 1-3 months, depending on how you take care of the car.