Learn Farsi: تو استادی؟ - Are you a professor?

It is attached directly to the end of the noun! The -i at the end of ostādi is a short verb ending (called an enclitic copula) that means 'you are'. So ostād means 'professor' and adding -i makes it 'you are a professor'.

How do we know this is a question without a word like 'are' at the beginning?

In Farsi, you do not need to change the word order or add a helper word to ask a yes or no question. You just raise the pitch of your voice at the end of the sentence. In writing, the Farsi question mark (؟) does the job. Both 'You are a professor.' and 'Are you a professor?' use the exact same words: to ostādi.

Is the pronoun to strictly necessary here?

No, you can easily drop it! Because the ending -i on ostādi already means 'you are', the word to (you) is completely optional. Saying just ostādi? is perfectly correct and very common. Including to just adds a little extra emphasis to 'you'.

I learned that adding -i to a word makes it an adjective or nationality, like Irani. How do I tell them apart?

You have to rely on context. Since this sentence starts with the pronoun to (you), it is a huge clue that the -i at the end of the sentence is the verb ending for 'you are'. If the sentence was about a third person, you would know the -i was just part of a descriptive adjective instead.