It's not something I've even needed to be involved with, at anything beyond casual interest, as we do not design any direct mains connected equipment.
Other people on here will likely have more detailed info.
But:
Don’t we need to draw more current when input voltage is low and vice versa (as P=V x I)? Otherwise we would need very big storage (Inductor or capacitor) to provide same power when AC input is low.
Yes - but: Firstly, think about a basic non-PFC supply - just a rectifier and capacitor input.
That will only take current for a relatively small part of each AC half cycle, when the rectifier output voltage exceeds the reservoir capacitor voltage; a short "recharge" several times higher than the load output current.
The capacitor has to provide the load for the rest of the half cycle.
If the PFC stage can provide current for a larger percentage of the cycle, the capacitor is needed for less of each half cycle so can presumably be smaller.
Also, the input current to the capacitor will be lower as it is available longer.
I'm not sure on the feedback part of things; from bits of info I remember, I believe there are both independent PFC stages and combined systems in use in different designs.
4) In some websites PFC has been described as a boost converter the role of which is to provide a constant voltage to the main flyback input (which should be , usually, couple of voltage more than maximum ac input). If so, then the same question comes again (relation of P, V & I).
That sounds about right, in concept.