Learn Spanish (Latin America): Para el martes, escribe el contrato profesional para el cliente. - By Tuesday, write the professional contract for the client.

No, they are doing two different jobs! The first one (Para el martes) sets a deadline, meaning "By Tuesday." The second one (para el cliente) shows the recipient of the contract, meaning "for the client." Both are key uses of para.

Could I use por instead of para for the deadline?

No. When talking about a deadline (having a task done by a certain day or time), you must always use para. If you used por, it would sound like you are writing the contract during Tuesday, rather than having it finished by then.

Who is being told to write the contract?

The verb escribe is an informal command, meaning it is an instruction given to someone you would call tú (you, informal). Regular affirmative informal commands look exactly like the "he/she" present tense form of the verb.

Why does the adjective come after the noun in contrato profesional?

In Spanish, descriptive adjectives like profesional almost always come after the noun they describe (contrato). So instead of "professional contract," the Spanish word order is literally "contract professional."