How can I properly model paper airplane physics?
I would model the energy flows between kinetic and potential energy. Start with a full bucket of potential energy plus the appropriate speed and drain kinetic energy away over time depending on flight speed.

To know how much energy is lost to drag, you need to model the drag using two components:


Friction drag, which grows with the square of airspeed, and
Induced drag, which drops with the square of airspeed.


If your drag is the sum of both, its minimum will be at some moderate speed. I plotted the drag components for a glider below, but since the physics are the same for a paper airplane, this plot should do for now.



The nonlinear behavior of the induced drag curve at low speed is due to flow separation, and something very similar will happen for a paper airplane. The important thing is: The drag curve has a minimum.

The energy loss over time is drag $D$ times speed $v$. This energy $E$ has to come from the reduction of height $h$ over time $t$:
$$\frac{\delta E}{\delta t} = \frac{\delta(m\cdot g\cdot h)}{\delta t} = D\cdot v = m\cdot g\cdot v_z$$
with $m$ the mass of the paper airplane, $g$ gravitational acceleration and $h$ its height above ground. Only $h$ changes over time, so derivation is easy and the derivative of $h$ is $v_z$, the vertical speed.

For picking a realistic drag it helps to rephrase the equation above by introducing lift $L = m\cdot g\cdot n_z$. $n_z$ is the load factor and is approximately one in straight flight. The ratio between lift and drag for a paper airplane is somewhere between 4 and 10 - just pick a number which results in a realistic simulation. To calculate the sink speed $v_z$ as a function of flight speed $v$ use this formula:
$$v_z = \frac{c_{D0}\cdot S\cdot v^3\cdot\rho}{m\cdot g\cdot 2} + \frac{m\cdot g\cdot 2}{v\cdot\pi\cdot b^2\cdot\rho}$$
S is the wing area of the paper airplane and $b$ its wing span. $\rho$ is air density, and for the zero-lift drag coefficient $c_{D0}$ you should pick a number which makes the paper airplane look realistic. Start maybe with 0.05.