From a military cargo pilot point of view, anytime you want the emergency equipment standing by you should declare an emergency. I have never heard of asking for the emergency equipment to be standing by without declaring an emergency. I understand the airlines have a different beast to deal with that being passeger comfort and hassle along with the perception a passenger may have of an airline after seeing lights and firetrucks surround the plane. All that possibly leading to lost customers. As far as waiting to declare an emergency, I have seen that on numerous occasions in the Air Force. Sometimes the situation does not dictate an immediate request and the pilot desires to continue and opt to declare when with approach control or center prior to arrival. This would occur when an immediate landing is not required and the pilot decides to contimue to a location with repair capability (Air Force Base with like aircraft) but wishes emergency equipment standing by. As far as paperwork, it's very minimal if any. You won't see a military aircraft (at least not a C-130) not declare an emergency to avoid paperwork. Hope that helps a bit.
P.S. I'm from Little Rock AFB, Arkansas and we coordiante with you guys (Memphis ATC) a lot during our training over Arkansas. Thanks for your work! We appreciate you guys working with us and accomodating some of our what may seem like weird requests.