I know that the Lagrangian $L$ is defined to be $T-V$, i.e. the difference between kinetic energy and potential energy. Also the Action $S$ is defined to be $\int Ldx$ and from this we can derive Newton's 2nd law of motion.
If we get Newton's second law out, does it mean that the formulation is correct? Couldn't it be just a coincidence?
Where do we derive these expressions for the Action and for the Lagrangian from?
Newton's second law, $\mathbf{F}_{net}=\dot{\mathbf{p}}$, is the definition of force. Lagrangian and action are defined to be $ T-V $ and $\int L\: \mathrm {d} t$ (and not $\mathrm {d} x $) respectively. You don't derive anything from anything here (however we can talk about how $ T $ and $ V $ come about).
This site derives the principle of least action from Newton's laws. http://www.damtp.cam.ac.uk/user/tong/dynamics/two.pdf
