Interesting question. Actually we have the energy observable (that is what is measured) and the Hamilton operator. If we measure the energy of a quantum system we should get a value from the operator's spectrum.
The measurement problem is really tricky and under debate, but energy is always associted to the hamiltonian.
It is derived. It doesn't always hold, because it really only applies to particles in free space. If there is a non-constant potential present, then you cannot necessarily even assign a single "wavelength" to the eigenstates (only a spectrum of them). For example, the Hydrogen atom's ground state is exponentially decaying with radius.