Is it necessary to re-wrap the house when we tear off the siding?
When repairing a small section of siding, it's a best practice to tape the holes and rips in the weather wrap (which you refer to as "paper") with the appropriate tape. But for a larger area, it's faster to simply replace the entire section of weather wrap and I would insist on it. Nail holes and tears compromise the effectiveness of the material, so I wouldn't recommend leaving it on when residing the house.