1. He lay on the floor.

2. He was lying on the floor.
[Are both okay? Which one is commonly used?]

3. He laid the box on the table.
4. He put the box on the table.
[Are both the same?]

5. He lied to her.
6. He told a lie to her.
[Ar both grammatical?]

All are grammatically correct, yes.

1. He lay on the floor.

2. He was lying on the floor.
[Thank you for your help. Does #2 mean state or action?]

The verb “lie” and all its forms are considered action verbs. Someone is doing something.

1. "He lay on the floor" is in the simple past tense, while "He was lying on the floor" is in the past continuous tense. Both are grammatically correct and can be used interchangeably depending on the context and emphasis.

2. Both sentences are similar in meaning but have slight differences. "He laid the box on the table" suggests that he placed the box on the table in the past, while "He put the box on the table" indicates he performed the action of putting the box on the table. In practice, however, many people use "put" and "laid" interchangeably.

3. Both sentences are grammatically correct, but they have different meanings. "He lied to her" means that he intentionally deceived her by providing false information. On the other hand, "He told a lie to her" means that he intentionally made a false statement to her. Therefore, both sentences are grammatical, but they convey slightly different nuances.