Is it possible to calculate the spontaneous decay rate of an atom's (for sake of "simplicity" hydrogen) excited state in QED, if so how and if not why not?
Quantum field theory isn't very good at describing bound states so we would not us it to calculate the lifetime of an atomic excited state. In any case there is in most cases no reason to use QFT because the non-relativistic calculation works very well. We might then use QFT to calculate the small corrections to the result we get from using standard QM.
The standard QM calculation is done using Fermi's golden rule to calculate transition probabilities. See for example How long does it take for an excited electron to return to ground state? or searching the site for Fermi's golen rule will find you many other related questions.