Here is a scene file that lets you launch and fly 5 different paper airplane designs:
paperAirplaneLaunch.ma (you'll find this at the bottom)

The airplanes are all made with nCloth and take advantage of the properties of lift and drag on the cloth. When you load the scene a heads up UI appears with several attributes. The first is airplane shape and lets you choose one of the 5 designs to fly. Here is number 5:

Number 3 is the record holder for long distance flight. I was surprised to find that after giving the nCloth version its first test flight it flew further than any of the other 5 designs.

To run the simulation simply playback. (note that due to a bug if you are running an XP64 version of Maya8.5 the playback will be very slow... the 32 bit version will be faster) You can set the launch speed and elevation before starting playback. Also you can rotate the plane with the rotate tool to push it at different angles of attack. The flaps can be adjusted while the plane is flying but they are sensitive to use( a joystick would be nice ).

Each plane has its own camera that follows it around( see this tutorial ). Note that you can even use Maya camera controls while the simulation is running to better follow the plane, however if the view gets far off you may need to reframe the plane. You can click on the button "Fixed Camera" to switch to a non-following camera.

It is possible to edit the planes. Scaling and rotation are fine, although if you want to edit cvs first select the "Edit Plane" button. When done editing select "Simulate Plane" before playing back.

Basically the "edit plane" button is just doing "nCloth: Display Input Mesh".

Real paper airplanes are folded in a way that makes them heavy and stiffer in parts. I simulated this by painting the mass per vertex to be greater where there would be more paper. This is very important for the flight behavior.

The launching was accomplished by accellerating a single face plane and constraining the paper airplane to it. The launch speed determines the point at which this constraint is broken by toggling the disable flag. I was able to share the constraint among all the airplanes by setting the "component relation" to "all to first" on the constaint. This connects all the constraint's nComponents to the first one, rather than connecting them all to each other.

The flaps were done by poly combining rotating flaps with the base plane meshes and merging verts before making nCloth. This was followed by "Edit nCloth: Connect Input Mesh to Rest Shape". The nCloth also needed high stretch and bend resistance to simulate paper. Needless to say, lift, drag, mass and gravity were important to get right for the final effect. Gravity is low in this scene, but I think it is more fun to see the planes in slow motion.

The following file shows how you can simulate a lot of planes very quickly. All the planes were first merged with poly combine before making nCloth.(there is only 1 nCloth in the scene). The playback should be pretty much realtime. The final animation has selfCollision enabled on the cloth which is important for the final look, but slows it down a little.