Both "well" and "good" can be used as adjectives, so both would be grammatically valid in the first sentence. When used as an adjective, "well" typically refers to good health, while "good" is more general and refers to something that is desirable and positive. Therefore when talking about people, "well" makes more sense. But when talking about something like the weather (in your second sentence), "good" makes more sense.

Personally in response to the question I would say "they are in good shape". I wouldn't say "They are good" because should I say it, I would mean "they are decent and nice people", thus having nothing to do with their physical well-being. Hope this makes sense.

The answer depends on the context. Are you asking about 'what is good and formal English?' or 'what do people say?'. In traditional/formal English 'well' is correct, for the reasons Julian explained. And that is why that is also the correct answer for an English test or assignment. In common usage, particularly among less educated people - good is probably a lot more common.