"Is that a boy or a girl?"

Just because it leaves out pointless words? :)

Az = that (or "the" if directly in front of a noun starting with a vowel)

Ez = this

That works as well, but it sounds much better with egy included.

Vagy is simply "or" in this sentence. (Vagy can also translate as "you are", so you're right to be wary.)

In copula sentences (sentences of the form "[Object] is [quality]") you don't translate "is" or "(they) are" into Hungarian. You just smoosh "object" and "quality" together:

• Ez fiú. - This (is) a boy.
• A ház magas. - The house (is) tall.
• Azok a féfiak nem orvosok. - Those men (are) not doctors.

If the subject is not a 3rd person ("he/she/it" or "they"), you use a form of "to be"/van, though:

• Fiú vagyok. - I am a boy.
• Magas vagy. - You are tall.
• Nem vagytok orvosok. - You are not doctors.