Learn Spanish (Latin America): ¿Tu profesor es argentino?

In Spanish, tu without an accent means "your", while tú with an accent means "you". Since this means "your teacher", we use the one without the accent.

Why do we use es instead of está?

We use the verb ser (which becomes es here) for nationalities and origins because they are seen as core traits of a person. We save estar for temporary states, feelings, or physical locations.

Why isn't argentino capitalized?

Unlike in English, nationalities and languages are written with lowercase letters in Spanish.

How would this change if the teacher was a woman?

You have to match the gender for both the noun and the adjective. It would become ¿Tu profesora es argentina?

The word order is literally "Your teacher is Argentinian?" Is that normal for Spanish questions?

Yes! In Spanish, you can often just use the exact same word order as a regular statement to ask a question. You just raise the pitch of your voice at the end and add the upside-down question mark at the beginning.