Learn French: Tu n'étudies pas dans le bureau ? - You don't study in the office?

Because the subject is Tu (you). For regular -er verbs like étudier (to study), the Tu form always ends in -es. Remember that you don't pronounce this final -s! The word sounds exactly the same as it does in the Je and Il/Elle forms.

Why is it n'étudies instead of ne étudies?

When the negative word ne comes right before a vowel (like the é in étudies), it drops its e and combines with the verb using an apostrophe. This makes the words easier to say together. The second negative word, pas, still goes after the verb.

Why is there no word for 'do' in this sentence?

French doesn't use helper words like 'do' or 'does' to make questions or negatives. The single verb étudies means 'study', 'are studying', or 'do study'. To make it a question here, you just raise the pitch of your voice at the end of the sentence.

Does bureau mean 'office' or 'desk'?

It can actually mean both! You have to use context to figure it out. Because this sentence uses dans (in), dans le bureau most likely means 'in the office' (the room). If you wanted to say 'at the desk', you would typically say au bureau.