What does "many times speed of light" really mean in this context? For a layman it's easy to draw wrong conclusions here. Is this just bad journalism or is there a good scientific explanation for this?
In curved spacetime, you can no longer compare velocities at different points in the straight-forward manner we use in flat spacetime. Thus the claim that recession velocities should not be considerer 'real' (as in relative) velocities, but rather rates of expansion of space. If you want to get at the former, you need to parallel transport the source's four-velocity along the light path to the observer. This should get you meaningful values below $c$ that correspond to the observed redshift (but I have to admit that I never went through these calculations myself).
Also note that superluminal expansion isn't really the defining characteristic of the inflationary epoch: Hubble's law relates recession velocities to distance, and if you move out far enough, the expansion in still superluminal today - there's not really any special significance to the speed of light in this particular context. Case in point: In principle, we should be able to observe galaxies with recession velocities of about $4c$ or so when their light got emitted.