Lagrangians and Hamiltonians

Why does physics work with an action?

What is the secret of extremizing a functional?

Sometimes for me it is a matter of mystery or faith that the action principle can indeed encapsulate all the physical content of a theory. The action: a single scalar, made out of scalars, without which, nowadays, no theory is really trusted too much.

What does it mean that a physical theory must have a Lagrangian descrition? It is clear that not all second order differential equations come from a Lagrangian. Therefore, it looks like physics selects a tiny subclasses of possible dynamics.