Learn Spanish (Latin America): Perdón, ¿usted es el doctor?

Usted is the formal way to say "you". We use it here to show respect, which is standard when speaking to a doctor, a stranger, or someone older than you. If you were asking a friend or someone you are very close to, you would use the informal tú.

Why does the formal "you" (usted) use the verb es instead of eres?

Even though usted means "you", it grammatically pairs with the "he/she/it" (third-person) forms of verbs. Because we use es for "he/she is", we also use es for "you (formal) are". If you were using the informal "you", it would be tú eres.

I learned before that we don't use words like "a" or "the" with professions. Why do we say el doctor here?

You're right that we usually drop the article when just stating a person's profession (like soy doctor for "I am a doctor"). However, in this sentence, the person isn't asking "are you a doctor in general?", but rather "are you the specific doctor?" (perhaps the one they have an appointment with). Because they specifically mean "the doctor", we need to use the word el.

Can I switch the order and ask ¿Es usted el doctor? instead?