頼む and 願う: requests and wishes
Compare requesting an action with wishing for an outcome.
Requesting or entrusting
頼む (たのむ) means asking someone to act or entrusting them with a task. 「友達に手伝いを頼みました」 says that the speaker asked a friend for help. The request alone does not establish acceptance.
「料理を頼みました」 can mean ordering food in a restaurant. 頼む covers requests for food or services as well as personal help.
A desired outcome
願う (ねがう) means wishing for something to happen. 「家族の幸せを願っています」 expresses a wish for the family's happiness, not a request assigning them a task.
「無事に帰れることを願っています」 expresses a hope that someone can return safely. It neither reports a completed return nor commands the person to return.
A request expression
Since 「お願いします」 is widely used for requests, 願う is not limited to private wishes. 「確認をお願いします」 requests a check, while 「成功を願っています」 expresses a wish for success. Read the object and situation together.
「頼まれた仕事を終えました」 means finishing the work one was asked to do. 頼まれた presents the recipient's perspective, not that of the person assigning the task.
Was the request accepted in 「頼みましたが、断られました」?
Show the answer and why
No. The speaker asked but was refused. Requesting and accepting are separate stages.