Classical method VS Hamiltonian method

Question

1. I'm very confuse with the method using Hamiltonian to derive the equation of the movement. In example I have, it's easier to derive the equation of the movement using classical method (namely 2nd law of Newton). Can someone give me an example where using Hamiltonian is much easier than classical method?

2. Also, is there a reason so that the Hamiltonian is the Legendre transform of the Lagrangian $T-V$ (where $T$ is the kinetic energy and $V$ the potential energy)? By the way, why is this the Lagrangian? All this looks a bit magic for me.