Does Paraphrasing have to involve an understanding of the mood?

No, a paraphrase usually does not reflect the mood of the original, unless it actually changes the sense. If the original is sarcastic, then the paraphrase should reflect the intent of the sarcastic utterance. But generally speaking mood is a product of the connotations of the Words, which is lost when the Words are changed.

E.g.

Don Juan: "Darling, I beg you to come away with me. A minute without the sight of you is a torture to me. I want you with me always."

Capt. Obvious: "I think he wants you to go with him."

OR

Col. Gestapo: "You will accompany me now, unless pain is something you enjoy."

Capt. Obvious: "I think he wants you to go with him."

Both accurate paraphrases of two statements with totally different moods.