1. He has lunch at 12:00.

2. He has lunch at 12:00 p.m.
3. He has lunch at 12.
4. He has lunch at 12 p.m.
5. He has lunch at noon.
(Are all grammatical?)

6. He goes back home at 15:30 p.m.
7. He goes back home at 15:30.
8. He goes back home at 3:30 p.m.
9. He goes back home at 3:30.
(Are all acceptable?)

Most of us don't use the 24-hour clock, so 6 and 7 would not be acceptable to a lot of Americans.

Correct.

But It would still be acceptable, just because many don't use does not mean that its not Acceptable. Its still correct.

I've not seen people use PM with the 24 hour clocks. Is that correct? I honestly do not know.

1. He has lunch at 12:00. - This sentence is grammatically correct. It uses the 24-hour clock format to indicate that he has lunch at exactly noon.

2. He has lunch at 12:00 p.m. - This sentence is also grammatically correct. It uses the 12-hour clock format with "p.m." to indicate that he has lunch at noon.

3. He has lunch at 12. - This sentence is grammatically correct, but it might be ambiguous because it does not specify whether it is referring to 12 a.m. (midnight) or 12 p.m. (noon).

4. He has lunch at 12 p.m. - This sentence is grammatically correct. It uses the 12-hour clock format with "p.m." to specify that he has lunch at noon.

5. He has lunch at noon. - This sentence is grammatically correct. "Noon" specifically refers to 12 p.m., so it is clear when he has lunch.

Regarding the second set of sentences:

6. He goes back home at 15:30 p.m. - This sentence is not grammatically correct. In the 24-hour clock format, there is no need to include "p.m." because the format itself indicates that it is in the afternoon. The correct format would be "15:30" without "p.m."

7. He goes back home at 15:30. - This sentence is grammatically correct. It uses the 24-hour clock format without "p.m." to indicate that he goes home at 15:30, which is equivalent to 3:30 p.m.

8. He goes back home at 3:30 p.m. - This sentence is grammatically correct. It uses the 12-hour clock format with "p.m." to specify that he goes home at 3:30 in the afternoon.

9. He goes back home at 3:30. - This sentence is grammatically correct. When "a.m." or "p.m." is not specified, it is usually understood that it refers to the afternoon. Therefore, "3:30" without "p.m." would indicate that he goes home at 3:30 p.m.