Writeacher, I urgently need you to check these sentences. Thank you.

1) They got him to give back the animal: He was persuaded to give back the animal. They had him give back the animal: He was requested (kindly invited) to give back the animal.
2) He forgot closing the door. He forgot that the door had been closed. I agree with you to go home early this evening.
3) You deserve to have a good score in this work. I hope to go on holiday to Greece this summer. Carl offered to help me. He offered me a helping hand (??). My mother promised (to me?) to go to the doctor. She invited me to her birthday party. Can I invite you to have dinner with me this evening?
4) Can I propose having dinner …. (?). I’m resisting being attacked by the enemy. He resents having to do this exercise. I have been told to go directly home.
5)I warned you not to trust him. I talked you out (??) Parking is not permitted here. It is not allowed parking the car here. You can’t stop me doing (from doing?) what I want.
6) He needs to go to the dentist.
He has an appointment with the dentist.
He has a dentist's appointment.
He needs to make an appointment with the dentist.
7) take/write an exam (do is a mistake?)
make a decision (take a decision is a mistake?

1. All are fine, but the last one is a bit odd. I would not use it.

2. He forgot that he had closed the door. He forgot that the door had been closed. He forgot he had closed the door.

I agree that you should go home early this evening. ("agree with you to go" is not a normal combination of words for this context)

3. You deserve to have a good score in this work. I hope to go on holiday to Greece this summer. Carl offered to help me. He offered me a helping hand. <~~All are OK.

My mother promised to go to the doctor.
OR
My mother promised me she would go to the doctor.

She invited me to her birthday party. Can I invite you to have dinner with me this evening? <~~All are OK.
The last sentence would be better like this: Would you have dinner with me?

4. Can I propose having dinner …?

I’m resisting the enemy's attack.

He resents having to do this exercise. I have been told to go directly home. <~~All are OK.

5)I warned you not to trust him.

I talked you out (??)<~~I don't know what you mean to say here.

Parking is not permitted here. <~~OK

Parking is not allowed here.

You can’t stop me from doing what I want.

6) He needs to go to the dentist.
He has an appointment with the dentist.
He has a dentist's appointment.
He needs to make an appointment with the dentist. <~~All are OK.

7) take/write an exam (do is a mistake?)
<~~The first two are right; "do" is not.

make a decision (take a decision is a mistake? <~~"make" is right; "take" is not.

1) The sentences are grammatically correct. However, if you want to check for clarity and style, I would suggest the following:

- They convinced him to give back the animal.
- They had him return the animal.
- He was kindly asked to give back the animal.

2) The sentences are grammatically correct. However, the last sentence could be rephrased for better clarity:

- I agree with you about going home early this evening.

3) The sentences are grammatically correct. However, there are a few suggestions for improvement:

- You deserve a good score on this work.
- I hope to visit Greece this summer.
- Carl offered to help me. He extended a helping hand.
- My mother promised to take me to the doctor.
- She invited me to her birthday party. Would you like to join us for dinner this evening?

4) The sentences are grammatically correct. However, there are a few suggestions for improvement:

- Can I suggest having dinner?
- I am resisting the enemy's attack.
- He resents having to do this exercise.
- I have been instructed to go directly home.

5) The sentences are grammatically correct. However, there are a few suggestions for improvement:

- I warned you not to trust him.
- I persuaded you not to trust him.
- It is not allowed to park the car here.
- You can't stop me from doing what I want.

6) The sentences are grammatically correct. However, there are a few suggestions for improvement:

- He needs to visit the dentist.
- He has an appointment with the dentist.
- He has a dental appointment.
- He needs to schedule an appointment with the dentist.

7) The sentences are grammatically correct. However, there are a few suggestions for improvement:

- take an exam (write an exam is also correct)
- make a decision (take a decision is less common, but acceptable)