Harmonics produced by LCC converters
The linked answer should answer it (you say you agree, I won't repeat everything in there). I'll add that there is no such thing as harmonics on the DC side. Harmonics are associated with harmonic signals, and DC is not harmonic.

Those are called DC link capacitors and their purpose is to store the energy for the next stage, such that the higher their value, the lesser the voltage fluctuations. There is a limit to the size of them, both in terms of physical realization, and as far as the control loops go (whose purpose is to maintain the voltage at a certain level).
 The fluctuations need to be kept low because, usually, the AC-DC-AC chain implies the usage of a switching inverter. These use PWM to deliver the AC, and they rely on the power supply to be delivered by the DC link capacitors. Due to the nature of the PWM, a varying supply means a varying amplitude, other than sinusoidal, which acts as an amplitude modulation, introducing not only harmonics, but risking causing sags that can damage the loads (and not only).
 Note that the DC links are not only capacitors, they can be inductors, too, since they, too, are reactive elements. In this case, the topology of the inverter changes, but the principle remains: having a large storage for the energy to maintain a steady delivery of voltage/current.

Point 1) shows what a nonlinear load that has no power factor correction (of any kind) does to the current. Since V=R*I, having those current peaks means greater voltage drops across the source which, in turn, can cause sags (sometimes potentially dangerous, e.g. in a weak grid). Improving the harmonic content of the current means less peaks which, in turn, means less sags, thus the voltage drops less. So, yes, power factor correction does improve voltage, too, though nothing is perfect: the switching regime and the parasitic inductances will cause additional filters, which, in turn, means more voltage drops. But, overall, the results can be much better than without, which is why there is a whole industry dealing with these matters.