1. He behaved most carefully.

2. He behaved the most carefully.
3. He behaved very carefully.

(Are they all grammatical? Does #1 mean #2 or #3? Thank you.)

All are grammatically correct, but #3 is the best of all. #1 is OK. If you use #2, you need to complete the comparison: "... the most carefully of all the students." (example)