I think the translation for "vediamo" is "we will see" in this example, as there would be no need for the word "domenica" otherwise. Sunday, we will see the cook's kitchen. It means that in the next few days, or very soon, so this would not require a future tense. It is unlikely to say it otherwise in English.
It can mean the noun 'kitchen' or the verb 'cooking/cooks' (when used with lei/lui/Lei) - Lui cucina - he cooks or he is cooking. But if you were talking about the cooking where cooking is used as a noun form I suspect it's different. Maybe (but don't know for sure cucinando since that's the present gerund form). If they use that form of it at all.
Hmm, apparently 'la cucina' can sometimes be translated as 'the cooking'. So technically, unless some native speaker knows why not, it is as valid as 'the kitchen'. But on the other hand, are you referring to the results of the cooking? Or the act of cooking? (The latter I think it would use a verb form with the cook).