Nature of motion between comoving observers; What is the common time that they agree on? This is a set of follow-up questions regarding this post. The following four queries are very closely related and needed to asked at the same place.
Question 1 Is it really possible to regard the comoving observers to be at rest w.r.t each other even though the physical distance between them is changing with time? 
Question 2 Since the double time derivative of the scale factor $a(t)$ i.e., $\ddot{a}\neq 0$, does this mean that the comoving observers are really non-inertial observers?
Question 3 Let us assume that the answer to Question 1 is 'yes' i.e. they can be assumed to be at rest w.r.t each other. Now, when one says that the comoving observers agree on an universal time (assuming synchronized) what kind of choice of time is being referred to here? Is it the proper time observed by any one of them? 
Question 4 If the answer to the Question 3 is 'yes', I would ask whether this proper time is synonymous with the term cosmic time (used to define a spacelike hypersurface on which the Universe is homogeneous and isotropic)?
 A: Question 1 Comoving observers are not at rest with each other because as you say their proper distance is changing. What you seem to mean is that in FRW-coordinates their position doesn't change.
Question 2 No, being in free fall a comoving observer defines an inertial frame locally.
Question 3/4 The comoving time coordinate of observers who perceive the universe isotropic designates the elapsed time since the Big Bang which is called 'cosmic time'.  
A: Some of your questions also arise in the case of static spacetime which is easier to understand. On the cosmic scale spacetime is not static, but I'll treat the static case first and then answer for the dynamic case.
First let's consider motion around the Earth in Newtonian physics, but introducing an accelerating reference frame.
We consider a set of balls or rocks or whatever falling towards Earth in the absence of air resistance. This is a case where spacetime is static. Let $(r,\theta,\phi)$ be ordinary spherical polar coordinates.  Let $r_{\rm ref}(r_0, t-t_0)$ be the radial distance (from Earth's centre) at time $t$ of a ball which was dropped from rest at $r_0$ at time $t_0$. Now define the coordinate $\chi = r-r_{\rm ref}(r_0, t-t_0)+r_0$. Then for each falling rock there is a $r_0,t_0$ such that that rock's worldline is described by $\chi = r_0 = $ const. This shows that we can find a coordinate system such that all these rocks in free-fall can be said to be at rest relative to this coordinate system, and therefore relative to one another if we choose to define distance and time using the coordinate system. (And even if I messed up the defn of $\chi$ in my example then the point would still apply to comoving coords in cosmic spacetime).
This answers your question 1, to which the answer is 'yes'. 
Question 2 is easy: if something is accelerating relative to some other object then it can easily be in free fall; this is the common experience of almost any situation of free fall motion. Such motion is inertial. The cosmic expansion is an example of free fall motion of a large number of objects, and consequently a dynamic change in the spacetime which they cause by their own gravitation.
Question 3. Yes, exactly right: it's the proper time observed by any one of them.
Question 4. Yes: synonymous.
