Why doesn't my flight aerodynamics maths work? Context:
For some context, I'm a game developer and I'm building a flight sim game. My goal is to have realistic flight physics -- not arcade physics.
I'm having issues with the maths -- it is not behaving how I would expect an aircraft to behave. Bear in mind, I'm no aerodynamicist!
I use a constant thrust directed forward (local), a constant weight force directed down (world), lift calculated with the below equations directed perpendicular to velocity, and drag directed opposite to velocity. I also have a down force provided by the elevators.
My plane is loosely based on an A320 using a wide variety of figures found online. The mass is 72,000 kg, wing span is about 35m, wing area is about 122m, engine thrust is 110,000N each.
My Maths So Far:
$$L = C_L \cdot \frac{\rho}{2} \cdot v^2 \cdot A$$
$$D = C_D \cdot \frac{\rho}{2} \cdot v^2 \cdot A$$
Source
$C_L$ is estimated with a table against angle of attack where $0^{\circ} = 0.5$, $5^{\circ} = 1.1$, $10^{\circ} = 1.45$ etc.
$C_D$ is estimated, where $C_{D_{min}} = 0.025$, $e=0.75$, and $AR = \frac{\text{Wing Span}^2}{\text{Wing Area}}$, as:
$$C_D = C_{D_{min}} + \frac{{C_L}^2}{\pi \cdot AR \cdot e}$$
Source
Issue:
When the plane accelerates it doesn't lift off the ground until about 300 knots. When it does so, it falls back down momentarily, bounces off the runway, and then climbs rapidly.
The elevator forces are clearly wrong. Before, I was just using a simple slider where I would manually select a force to apply to the elevator ranging between -300,000N to 300,000N. I then tried using the lift equation with an estimation of the wing size and area, but the force was much too strong. It produced too much torque and the plane would spin almost on the spot. I also wasn't sure how to control the lift equation with user input.
The drag force also don't work properly. Even when I reduce the thrust to zero, the drag force produced is so minuscule it would take forever to decelerate the plane.
This is a screenshot of my airplane on the runway. At the time of taking the screenshot, the aircraft was travelling at 212 knots, it was producing 444,000N of lift, and drag was 16,000N. The blue square represents the resultant force.

With that all said, here are some specific questions:


*

*Can anyone identify something I've done wrong? Are the equations/constants/applications etc. ok?

*Will a real plane lift off the ground on it's own after reaching a certain speed (without elevator input from the pilot)?

*Is there an equation for the horizontal stabilizers? Is it just the regular lift equation but directed down?

*How do flaps, elevators, ailerons etc. affect the lift equation -- how can I accurately model this with maths?

*Where are the lift forces applied? Is it always applied about the center of mass even when turning/climbing?

 A: If unity handles forces for you, then you must apply each force at the point on the plane where it is generated. For example for your wings apply the lift force of each wing at the center of mass of that wing. Do not apply the force at the center of mass of the whole airplane. That is because unity can only calculate the correct torque that way.
One game which does a good job at such a simulation is kerbal space program. I suggest you read this tutorial on airplane design for that game. Basically the center of lift (CoL) must be above the center of mass (CoM). If you have moveable control surfaces on your plane, they should to be placed such that the CoL is slightly behind the CoM for stability, but the CoL moves slightly ahead of CoM when you move the control surfaces. The plane then only takes off when pulling up.
To calculate the CoL you need to take the sum of all lift forces on the body and then calculate the point relative to the CoM at which the total lift force would generate the same torque as all the lift forces together. But you can also just play around with the position of your wings until you get a stable aircraft.
A: 
I found this equations:
$$\sum{F}_x=m\,a=F-R-W\tag 1$$
$$\sum{F}_y=N+A-m\,g=0\tag 2$$
where:
$F$ thrust force
$R=\mu\,N$ rolling resistance force
$W=\frac{1}{2}c_W\,\rho\,S\,v^2$ air resistance force
$A=\frac{1}{2}c_A\,\rho\,S\,v^2$ lift force
$S$ wing area
From equation (2) 
$$N=m\,g-A=m\,g-\frac{1}{2}c_A\,\rho\,S\,v^2$$
so for  $N=0$ we get:
$$v_S^2=\frac{2m\,g}{c_{AS}\,\rho\,S}$$
where $c_{AS} < c_A$
from equation (1) you get:
$$m\,a=F-R-W=F-\mu\left(m\,g-\frac{1}{2}c_A\,\rho\,S\,v^2\right)
-\frac{1}{2}c_W\,\rho\,S\,v^2$$
after some calculation and  with $c_R=c_W-\mu\,c_A$
 you get:
$$a(v)=\frac{c_R\,\rho\,S}{2m}\left(\underbrace{2\frac{F-\mu\,m\,g}{c_R\,\rho\,S}}_{v_E^2}-v^2\right)$$
the plane can only takeoff when $v_E > v_S$
the take-off distance is:
$$s_S=\int_{0}^{v_S}\,\frac{v\,dv}{a(v)}=-\frac{m}{c_R\,\rho\,S}\ln\left(1-\frac{v_S^2}{v_E^2}\right)$$
For Airbus A340 with:
$F=$ 600 [kN]
$m=$ 275 [t]
$S=362$ $[m^2]$
$\mu=0.04$
$c_{AS}=1.9\,,c_A=1.5$
$c_A/c_W=5$
$\rho=1.21 \quad [kg/m^3]$
you get:
$v_S=290 \quad [km/h]$
$v_E=348\quad [km/h]$
and
$s_S=3085\quad [m]$
A: It is not clear how you take into account the rotation of the airplane around the transverse horizontal axis. For example, when the airplane takes off, the thrust is not directed horizontally anymore. You should take into account the points where the forces are applied: the lift is applied mostly to the wings, the weight is applied to the center of mass, etc.
A: This is going to take a lot more than two equations.  If I was coding this problem, I would start with a free-body diagram of the airplane, and draw all forces acting on it.  The reference plane (rather than reference frame) would be level ground.  Forces would have to be broken into horizontal and vertical components relative to this plane.
In addition, I would identify the center of gravity, and determine the torque of the airplane around the three known rotation axes, based on the forces that have been identified.  The rotation rate would depend on the moment of inertia of the airplane in each of these axes, which may not be all that easy to get, since the airplane is not a "convenient" shape and it's weight distribution around those axes may not be ideal.  In addition, there will be drag associated with rotation, and you will have a different drag coefficient associated with each axis of rotation.
This is NOT a simple problem.  It may be better to search for the most realistic arcade physics example you can find, do some research on that example, and implement a solution that uses your particular airplane based on that example.
