1. Forgive me. I didn't know that.

2. Excuse me. I didn't see you there.
3. Pardon me. I made a mistake.
(Are they all grammatical, showing 'forgiveness'?)

4. I swear that I will never leave you.
5. I swear to God that he will never leave you.
6. I swear in the name of
God that she will never leave you.
7. I bet he is honest.
Definitely he is honest.
8. I am sure he is honest.
(Are they all grammatical?)

1. Forgive me. I didn't know that.

2. Excuse me. I didn't see you there.
3. Pardon me. I made a mistake.
(Are they all grammatical, showing 'forgiveness'?)

4. I swear that I will never leave you.
5. I swear to God that he will never leave you.
6. I swear in the name of
God that she will never leave you.
7. I bet he is honest.
8. Definitely he is honest.
9. I am sure he is honest.
(Are they all grammatical?)

All are fine.

Yes, all of the sentences you provided are grammatically correct.

1. Forgive me. I didn't know that.
2. Excuse me. I didn't see you there.
3. Pardon me. I made a mistake.

These sentences express apologies or seeking forgiveness for a mistake or lack of knowledge. They are all grammatically correct and convey the desired sense of seeking forgiveness.

4. I swear that I will never leave you.
5. I swear to God that he will never leave you.
6. I swear in the name of God that she will never leave you.

These sentences express a strong commitment or promise. While they are grammatically correct, the usage of "swear" in sentences 5 and 6 is more emphatic since it involves invoking the name of God. It is worth noting that the phrase "swearing to God" is a personal choice and may carry religious connotations.

7. I bet he is honest.
Definitely, he is honest.

These sentences express a level of confidence or belief in someone's honesty. Sentence 7 uses the phrase "I bet" as a casual way to express confidence, while sentence 8 uses the phrase "I am sure" to emphasize a higher level of certainty. Both sentences are grammatically correct, though sentence 8 conveys a stronger sense of confidence.

Overall, all of the sentences you provided are grammatically correct and effectively convey their intended meanings.