How to start deriving longitudinal equations of motion for an aircraft?
Note: Answer in progress!

Part 1 
(Unfortunately I'm only familiar with #1 and #3 at the moment, not #2)

(Footnote: This might be a bit simpler than your case, but hopefully it you'll be able to fill in the remaining gaps)



From this, you can some the forces up according the direction of the velocity or the lift  vector. Doing this horizontally, you get equation 1, and likewise for vertical direction your equation 3.

To make this simpler to handle, we use small angle approximation consider $\cos(0)=1$ and $\sin(0)=0$. This simplifies down to:

$$V:0=T-D$$

$$L:0=mg-L$$

(i.e. thrust equal to drag, lift equal to weight)



Part 2: 

This is basically the equations of a kinetic diagram of the Free Body Diagram above, where there can be a change either in airspeed of altitude. What your second equation says is that excess thrust (T-D, Thrust-Drag) can: 


be used to increase altitude: $m\times{g}\times\sin(y)$


and/ or 


be used to increase airspeed: $m\times{v}$