que coûte le manteau ? is a formal question: what does the coat cost?
ce que coûte le manteau? is not a true question: what the coat costs? - you would say it to repeat someone's question asked to you, maybe to be sure you have well understood, like "you want to know what the coat costs? - tu veux savoir ce que coûte le manteau ?
in addition, you cannot use "qui" which is exclusively subject: in this sentence, the subject is "manteau"