Suddenly, a rainbow appeared in the sky. Is sky an adverb because it tells where the rainbow appeared?

also, The hard working plumber was early. Was early is the verb phrase. I have trouble labeling it linking or action verb because it seems like both to me. Can you help??

No. Sky is a noun, the object of the preposition, in. The phrase, "in the sky," is an adverb phrase.

The word, "early," is not part of the verb. It's an adjective, following the linking verb and describing "plumber."