What's the motivation behind the action principle?
Why does the action principle lead to Newtonian law?
If Newton's law of motion is more fundamental so why doesn't one derive Lagrangians and Hamilton principle from it?
Also does all Lagrangians obey $L=T-V$?
I think that it's related to the fact that the kinetic energy of the particle at all points on the path or it's travel time is as small as possible?
If so, How can we derive the principle of least action from this fact in detail?
The motivation is that it works. It was a major discovery that this unifies mechanics (in the absence of friction).
It doesn't lead to newton's law, as it is far mor general than the latter, but applied to the problems Newton was interested, it implies these laws.
Action is more fundamental than newton's laws. The latter are more intutive, though, hence were found much earlier in history and can be taught much earlier in education.
No. This is only the form of Lagrangians for $n$-particle systems in Cartesian coordinates - the stuff with which one begins in theoretical mechanics.
What is related? Minimal travel time is a vartiational principle different from the action principle and gives the same answers only in very special cases.
is not applicable.