Is there any reason why "watch" isn't correct as an alternative to "see"?
To see something and to actively look at (watch) it are two different things. If Duolingo suggests "watch" as a translation for "vedere", you should report it so that the devs can improve it.
to see = vedere
to watch = guardare
Probably the same reason ( whichever that may be) "we look at" is also not accepted as an alternative translation of "vediamo"...
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.
do I have to translate the sentence in the exact order that it is written? To say, "We see the cook's kitchen on Sunday" is correct, and it avoids the confusion between "domenica" and "la domenica" (habitual action)
why isn't "let's see" acceptable? the imperative noi ends in - iamo and translates to "let's _"
On Sunday we SHALL see the kitchen of the cook IS correct. And not the proposition of the authors: we will... !!!!!
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).