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 <u>performed</u>.
2) The rabbit appeared out of the hat.
Same explanation as the question 1. <u>Appeared</u> 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 <u>tastes</u> is the linking verb which connects the adjective and the subject.
4) They stayed best friends forever.
The linking verb here is <u>stayed</u>. (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.)