1) Possessive pronouns are usually not used with body parts in Czech (and many other languages, including French). Arguably "what do you have in the mouth" doesn't really make sense in English, and at best would be understood as "what do you have in your mouth ?"
2) ústa is a plurale tantum, which means it's a noun which is only used in the plural, but with a singular meaning (like "glasses" or "pants" in English).
Regarding possessive pronouns in sentences about body parts, I would say that in my neck of the English speaking world, the pronoun, though often used, is optional - "What do you have in the mouth?" (or "what have you got in the mouth?") is a used form of this question, especially to a small child!