1. He is at the back of the door. [incorrect]

2. He is in back of the door. [correct]
2-2. He is in front of the door. [correct]
3. He is behind the door. [correct]
4. He is in the back of the door. [incorrect]
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I'd like to arrange the use of 'front' or back'. Is my explanation in [ ] grammatically correct?

5. He is in back of the car.
[This means he is behind the car. He is out of the car.]
6. He is in front of the car.
[This means he is before the car. He is out of the car. He is not inside the car.]
7. He is in the back of the car.
[He is in the back part of the car. He is now inside the car.]
8. He is at the back of the car.
[He is at the back part of the car. He is inside the car. He is not out of the car.]
9. He is behind the car.
[He is out of the car. He is not inside the car.]
10. He is before the car.
[He is not inside the car. He is out of the car.]
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Is my explanation in [ ] grammatically correct?

Yes, I think you are correct. :)