There is no simple equation for how a paper airplane flies like there is for a simple projectile because the airplane can interact with the air in complicated ways.
The physics of a paper airplane is described by Newton's laws of motion. These laws apply to both the airplane and the air it travels through. The plane is acted on by a constant gravitational force and by contact forces with the air, especially drag and lift.
The nature of the force between the air and the plane can be quite complicated, and requires an extremely detailed analysis for accurate simulation. For example, by constructing the plane slightly differently, you can make it fly faster, slower, further, curve left or right, or bob up and down.
The basic physical ideas are those of fluid dynamics and the basic equation involved is the Navier-Stokes equation. Modeling something like an airplane accurately is mostly the domain of expertise of aeronautical engineers.
To make a simple model for a game, you might want to start with a simple constant gravity force, a drag force proportional to the square of the velocity, and a lift force also proportional to the square of velocity (which comes from here), and then play around with the parameters until you find something pleasing to your eye.
I assume you are going to program this. First you need to write the vector differential equations of motion. The derivative of position is velocity, and the derivative of velocity is acceleration. Acceleration is force divided by mass. Force is the sum of a number of components, mainly gravity, lift, drag, and thrust.
Then you just integrate those with any ODE solver you like. The simplest is Euler. If you want more accuracy, you can use Runge-Kutta. You probably don't need a stiff solver such as Gear or Adams-Moulton.
So if the motion of the aircraft is along a straight line with wings level at constant speed, then the lift vector is equal and opposite to the gravity force vector.
The way you turn it is to bank the wings to an angle. For example, if you bank the wings 45 degrees to the left, that tilts the lift vector 45 degrees to the left, so it's vertical component is .707 of what it was, and .707 of it is directed to the left. The sideways force causes the path to be a circular arc. The reduced vertical force causes the aircraft to start descending.
To compensate for that tendency to descend, the pilot increases the magnitude of the lift vector by adding back pressure on the control yoke.
You'll notice that you feel a little heavier in the turn.
The increased lift results in increased drag, so the pilot increases engine power to maintain speed.
Then when the aircraft has travelled far enough around the circular arc to be headed in the desired direction, the pilot levels the wings, releases the back pressue, and reduces the engine power. If he doesn't, the aircraft will start to climb.
You'll notice this the next time you fly.
That's how turning works. Now I'll tell you how straight flight works.
The lift and drag you get from the wing depends on two things, speed and angle of attack.
If you reduce speed, but want to stay in level flight, you need more angle of attack to get the same lift. So the way a pilot slows down a plane is to reduce power and then gradually pull the nose up by applying back pressure on the yoke.
Since it would be tiring to continue to apply back pressure when flying slowly, the pilot has a "third hand", the trim wheel. Rolling that wheel backward applies back pressure on the yoke so the pilot doesn't have to hold it.
In fact, the primary speed control of the airplane is the trim wheel.
The power doesn't actually control speed, it controls whether the aircraft ascends or descends at the speed it is travelling.
One last point is balance. If you take a plane on the ground, and place a jack under each wing at the center of lift and hoist it up, its nose will fall toward the ground.
It's center of gravity is forward of the center of lift.
So when it's flying, as the main wing is pushing up, the tail is pushing down.
The moment between the two is what keeps the nose from dropping.
This is very important for safety, because it stabilizes the speed.
If the speed decreases, that moment decreases, and the nose drops, causing the aircraft to start to go "downhill".
Since it is going downhill, it picks up speed.
Conversely if it's speed increases, it starts to go uphill, causing its speed to decrease.
If the aircraft is loaded too far aft, its speed and its up/down directional stability is lost.
In fact, it is possible to get nosed up, and then go into a backslide all the way to the ground!
Best Answer
Landing - stress is highest on the pilot, because he/she can't afford to be too fast/too slow, or too high/too low.
Takeoff - stress is highest on the engines. They are at full power and have to hoist a heavy aircraft as high as possible in as short a time as possible.
Cruise - stress is highest on the airframe when encountering up/down drafts at cruising speed (bumpy air). Look at it this way, every plane in level flight, for a specific weight and configuration (flaps, etc.) has a stall speed. That is the slowest speed at which it can support its own weight. If it is going twice that speed, it can support four times its own weight (and the passengers will feel 4G, and it can go negative as well). If it is going three times that speed, in principal its wings could support nine times its own weight if they were strong enough. Typically they are not strong enough, and they will break instead, if they encounter a strong enough updraft or downdraft. That is why every aircraft has a particular speed, a fairly low speed, called "maneuvering speed", that they slow down to if they stupidly stumble into a storm cell. At that speed, there is no amount of up or down draft that can cause structural damage. Technically, it's the speed at which maximum control deflections cannot cause structural damage. Remember the accident in Far Rockaway NY? The pilot tore off the tail fin by stomping the rudder pedals too hard from one side to the other.
By contrast, military fighters and aerobatic stunt planes are built for high-G turns (12G is possible). You can see, since available lift is proportional to velocity squared, it is not at all difficult to go fast enough to get that kind of lift.
Also, don't forget a jetliner is a pressurized air bottle, for high-altitude flight with passenger comfort. It cycles from un-pressurized to pressurized every time it climbs to cruise altitude and back. This has been known to cause metal fatigue cracks, resulting in some accidents.