Learn Spanish (Latin America): ¿Hay alguien en el cuarto vacío?

It means both! In English, we usually switch from "someone" to "anyone" when asking a question. Spanish doesn't make this switch, so you just use alguien for both.

How would I answer this question by saying "No, there isn't anyone"?

You would use nadie (nobody/not anyone), which is the exact opposite of alguien. Because Spanish requires double negatives, you would say No hay nadie (literally, "There is not nobody").

Since alguien refers to a person, why don't we use the "personal a" before it?

While alguien usually needs the "personal a" when it receives the action of a verb (like ¿Ves a alguien? meaning "Do you see someone?"), the verb hay is a big exception. It never takes the "personal a", even when pointing out people.

What is the difference between cuarto and habitación? Can I use either one here?

Yes, you can use either! Both mean "room." Cuarto is very common for general rooms or bedrooms in a house. Habitación is also widely used, though it can sound slightly more formal and is the standard word for a hotel room.