1. She looks happy.

2. She looks like a professor.
3. She looks like she is a professor.
[Are they all grammatical?]

1. She looks happy.

2. She looks like a professor.
3. She looks like she is a professor.
4. She looks as she is a professor.
5. She looks as if she is a professor.
6. She looks as if she were a professor.
7. She looks as if she was a professor.
[Are they all grammatical?]

These are correct: 1, 2, 6.

Sometimes 5 is used.

Yes, all three sentences are grammatically correct. They are different in terms of the information they convey.

1. "She looks happy": This sentence describes the person's appearance and suggests that the person appears to be in a state of happiness. It focuses on the person's expression or demeanor without providing any additional context.

2. "She looks like a professor": This sentence compares the person's appearance to that of a typical professor. It implies that the person possesses certain qualities or features commonly associated with professors.

3. "She looks like she is a professor": This sentence not only suggests that the person's appearance is similar to that of a professor but also implies that the person may actually be a professor. It emphasizes a likelihood or assumption about the person's occupation based on their appearance.

To determine if a sentence is grammatically correct, you can examine the structure, word choice, and syntax. In these examples, all three sentences are grammatically sound.