It is more or less caused by time dilation, yes.
Even in special relativity, with no spacetime curvature, there's no limit on how quickly two things can recede in terms of proper time. If two objects start at rest and then head off in opposite directions at $.99c$ for one second as measured by onboard clocks and then decelerate to their original speeds, they'll be about 14 light seconds apart at the end.
You can make a special-relativistic toy big bang model by having a bunch of objects move inertially away from a common starting point at many different speeds. They all recede from each other at less than $c$ if you calculate their speeds as $dx/dt$ where $x$ and $t$ are inertial coordinates. But if you calculate separation per proper time, there's no upper limit. This notion of "separation per proper time" is just not the kind of speed that is limited to $c$ in special (or general) relativity.
This toy model is actually a special case of the standard cosmological model (FLRW); it's the zero-density or zero-gravity limit, sometimes called the Milne cosmology, and it has "superluminal" recession speeds in the sense normally used in cosmology.
if the former is true,
(it is)
then no two particles in universe ever moved relative to each other FTL, right?
Locally, there is no superluminal relative motion. On a larger scale, it generally doesn't make sense to compare speeds because of spacetime curvature. But "superluminal" expansion at the cosmological scale has nothing to do with curvature as such, contrary to popular belief. In the flat-spacetime limit (the Milne cosmology), the "superluminal" cosmological expansion speeds are still there, even though there's also a well-defined special-relativistic global speed (which is different, and never exceeds $c$).
