In general relativity a reference frame need not be inertial. 

The universe does have a preferred reference frame. It is the one in which the CMB radiation is isotropic to first approximation. This is also the frame in which velocities of galaxies relative to the frame are zero on average. In cosmology it is called the "co-moving frame" and observers at rest in this frame are called "fundamental observers". Such observers find the CMB to be isotropic, and at any given cosmic time they also all agree on its temperature. But these observers are in motion relative to one another in the following sense: the length of a rod stretched between any pair of them changes with cosmic time. (The 'rod' here being millions of lightyears long so not a practical thing, just a thought-experiment). So your point 2 is correct.

None of the above breaks the principle of relativity, because for the purpose of mathematical analysis one can always adopt other frames if one wishes, and the Einstein field equation will still have the same mathematical form.

Finally, all this has nothing to do with any early period of cosmic inflation, because even if there had not been any such early period (and the evidence is ambiguous) then the above would still be true. 

The smooth nature of the cosmic expansion gives an example of the fact that the early conditions of the universe were very special, and this remains a puzzle. Inflation does not solve this puzzle because it too requires special initial conditions.