I suggest you rip that sheet off and replace it with gib. You'll get a much better finish but also the opportunity to inspect the inside of the wall and insulate - even though it's only a small section. Alternatively clean it up as best you can then apply a layer of paint-able wall paper (I think there's a special and cheap type for this exact purpose - ask your store). For your door frame sand it back till that painted edge in your photo is clean and re-paint.
Edit - for the gap (second photo) put gib along it.