The correct answer is “do”.
“Do” is the most common verb in the English language, and it can be used in a variety of ways. In this case, it is used as a simple present tense verb to express a plan or intention. The sentence “Let’s do this crossword puzzle together” means that we are going to start working on the puzzle together.
The other options are not as appropriate in this context. “Doing” is a present participle, which is used to form the present progressive tense. It would not make sense to say “Let’s doing this crossword puzzle together” because it does not express a plan or intention. “Did” is the past tense form of “do”, and it would not make sense to use it in this context because we are talking about something that we are going to do in the future. “To do” is the infinitive form of “do”, and it would not make sense to use it in this context because we are not talking about a general action but about a specific action that we are going to do together.