1) The actors performed a wonderful play.
Action verbs are verbs that express an action that the subject of the sentence is doing. In this sentence the action verb is performed.
2) The rabbit appeared out of the hat.
Same explanation as the question 1. Appeared is the action verb that describes what the rabbit is doing.
3) The sauce tastes too salty.
A linking verb is a verb that links the adjective that describes the subject together. In this sentence tastes is the linking verb which connects the adjective and the subject.
4) They stayed best friends forever.
The linking verb here is stayed. (A little disclaimer on this answer, I'm not 100% sure on this one, but I do believe this is the correct answer. Apologies if it is not.)