QED and QCD are specific quantum field theories. As a subject, quantum field theory is a subset of quantum mechanics: quantum field theory is the quantum mechanics of fields. QED is the quantum mechanics of the electromagnetic field. QCD is the quantum mechanics of the color field.
If you define electrodynamics to be the study of matter with the electromagnetic field (i.e. with photons), then QED is simply a fully quantum-mechanical approach to that study. In that sense, the "standard QM rules" apply since what you're doing is simply applying QM to yet another problem.
One caveat to this discussion is that the Born interpretation of the wavefunction becomes a bit shaky. The reason is that in quantum field theory (QFT), particle number is not necessarily conserved. This is of little concern because what we would normally call a wavefunction function in single-particle QM yields calculable results in QFT for things we actually measure like scattering experiments, etc.
Quantum mechanics deals with single particle systems, which have a small number of degrees of freedom (think observables.) For example, a single particle has associated observables x and p. A quantum field is the quantum mechanical version of the classical field, which describes (maybe infinitely) many degrees of freedom. What makes the quantum field quantum is that these observables are quantized. The observables are particles, making quantum field theory a multiparticle theory: the field describes the creation and destruction of quanta as a function of spacetime.
In QFT, the field operator becomes the dynamical quantity of interest, rather than the single particle wavefunction.