They are all adjectives because they describe things, not actions.
Taking the first one: "That question remains mysterious". What word in this sentence does "mysterious" describe? It is the question which is mysterious -- it was mysterious in the past and it is mysterious in the present.
If you use an adverb -- "That question remains mysteriously" -- it changes the mean. "Mysteriously" now describes the verb; the sentence means that the question remains, and it does so in a mysterious manner. We expected the question to go away, but it hasn't, and it is this which mystifies us. (The question itself is a normal question, and it isn't mysterious at all.)
(This word order, though, would be a bit unnatural. We would normally say: "Mysteriously, that question remains" or: "That question mysteriously remains".)
If you are describing an action, use an adverb.
But if you are describing a thing, use an adjective.
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