A Photon is a quantum of energy typically emmited by one Electron and absorbed by another.
The electron loses energy/momentum when "emitting the virtual photon" and therefore must be in a "lower orbit" until it reabsorbs said photon. As it is doing so all the time, its average position is nearer to the nucleus. I must say that this is just a pretty picture for experimentalists/laymen who do not understand QFT. Virtual particles are a mathematical concept which describes the effect of what is happening on the quantum level (up to some order), but not the thing itself. In the end there is no easy way to imagine what happens b/c during an interaction the particle picture (asymptotic states) breaks down and the, for us, unintuitive wave nature of the "particles" takes over. The electron field kind of interacts with itself and in doing so produces higher order effects like the one you are thinking about.
The self-energy is valid for any electron (or any particle w/ charge generally). But if the electron isn't bound, we can't observe it as a "lower orbit". So, it happens everywhere. I think it should be observable theoretically in any bound electron, but it is a very small correction (hyperfine structure). I would imagine that for most systems (eg bigger atoms, molecules and solid-state systems) our theoretical predictions are not precise enough (many body problem) for this correction to be measurable. –
Self-energy is only appearing second order in alpha in perturbation theory. At all energies we can achieve, alpha is very small (1/137). At very high energies the effect of the self-energy would become more important.
our calculations/models from which the not yet corrected values come are approximating more things, s.t. the uncertainty is much bigger than this correction. The hydrogen atom is a two-body problem and analytically solvable. Helium already needs some approximations to be made/numerical calculations to be solved.