Learn Farsi: آیا تو می‌خواهی در این دانشگاه درس بخوانی؟

Because it follows the verb می‌خواهی (you want). In Farsi, when you say you want to do an action, that second verb must take the present subjunctive form. For most verbs, this means replacing the continuous می prefix with the subjunctive ب prefix. So, می‌خوانی (you read/study) becomes بخوانی (that you read/study).

Why are there two words for the verb 'study' here: درس بخوانی?

The Farsi verb for 'to study' is a compound verb: درس خواندن (which literally translates to 'lesson reading'). When you need to use the subjunctive form after a verb like 'to want', you only apply the prefix change to the action part of the verb. The noun part (درس) stays exactly the same, giving us درس بخوانی.

Since the verbs already end in the 'you' suffix, is the word تو necessary?

It is not strictly necessary. Because both verbs in this sentence already end with the singular informal 'you' suffix (ی), the meaning is completely clear without the pronoun. Including تو just adds a little extra emphasis, like asking 'Do you want to study at this university?'

Do I always have to start a yes/no question with آیا?

No, you can often leave it out! The word آیا is mostly used in formal or written Farsi as a clear flag that a yes/no question is coming. In everyday spoken Farsi, people usually drop it and just raise the pitch of their voice at the end of the sentence to show they are asking a question.