It would be GatavYAT if the stress was on the last syllable. Since in this case the middle syllable is stressed, it's technically "GaTOVyet," the last syllable of which is often incredibly soft and sounds like an "i" to non-Russian ears
As a Russian (teacher of this account)I hear wrong pronunciation here. It should be "Gatoviat" as in они готовят where я is pronounced as "ya" or "ia" but not as "i".
No, it more generally means preparing something or getting ready. What it means should be clear from the context of the conversation. In this exercise, everyone is preparing some kind of food, so it is safe to assume that this sentence is also about preparing food.
Is there something wrong with the translation: "Yes, they are currently cooking."?