When all goes as it should the information gets passed automatically and a strip is generated before the actual hand off happens. The receiving controller won't take the hand off on an airplane he has no idea about. There use to be a computer system in each center which would receive information from the approaches and towers in their control area as well as other centers. It then decides what sector, approach, tower and/or other center needs that information and sends it there. Now it maybe consolidated somewhat, overall system works the same.
I haven't ever been to a true FAA TRACON so I don't know how many printers the normally have. The approach control I worked when I was in the military only had one printer. The printer was hooked into our FDP computer which received data our host center. When the center's computer had a flight plan that would enter our airspace it would send the info to our FDP computer which would print out the strip. This happened usually 30 mins from the time it was estimated to touch the boundary of our airspace. It was the clearance/flight data controllers job to take each strip and take to the position in the control room that would need it. When a plane passed between control positions inside the facility the strip was physically passed with it. If we had a plane that was leaving our airspace our FPD would send the data to the center's computer and the center's computer would send the data to whoever would need to the information and their FDP would print out the strip, again, about 30 before it hit their boundary.
At a small to medium sized tower there would only be one printer. As the planes moves from taxing to waiting for take off they would get handed to the local controller from the ground controller. I have no idea how it works at a large complex tower like JFK, ATL, DFW, etc. Some of these airports have two local controls some even have two towers.
Most centers don't have strips anymore. They use URET which displays and records the same information electroncially I haven't even seen it yet so can't really describe it, but the back end is probably mostly the same.