Doesn't dating the universe violate the concept of spacetime's inseparability? It would seem that measuring an age of the universe from the big bang requires separating spacetime into a 3D coordinate system and a time track.  I fail to understand why it is appropriate to take the reference of a comoving observer as being stationary to the cosmic microwave background when this seems to be a choice based upon our subsequent detection of that entity.
 A: You're quite correct that measuring time from the Big Bang does separate spacetime into a time bit and a space bit, but this isn't arbitrary.
When we want to describe the universe around us we need to choose some coordinate system that we can use to record physical quantities. Whatever coordinate system we choose will have one coordinate that behaves like time and three that behave like space. General relativity tells us that any coordinate system can be used, so there is no unique way to split our coordinates between time and space. However some coordinate systems are more convenient than others and it's often the case that one coordinate system is obviously easier to work with than others.
Another basic principle of relativity is that locally spacetime always looks like Minkowski space i.e. regular flat spacetime. And an obvious choice of coordinates for flat spacetime are the Cartesian coordinates $(t, x, y, z)$ with the origin centred on ourself. By choosing these coordinates we are choosing to split spacetime up into a $t$ coordinate with a family of spatial hypersurfaces for each value of $t$, but this is a natural splitting that's intuitively easy to understand. Even this simple splitting isn't unique. Effects like time dilation happen because observers moving relative to us split their coordinates in a different way and their time coordinate doesn't match ours.
Anyhow, it turns out that for describing the expanding universe we can make a choice of coordinates that is very close to the simple coordinates used in flat spacetime. We call these comoving coordinates. The metric for flat spacetime in Cartesian coordinates is:
$$ ds^2 = -c^2dt^2 + dx^2 + dy^2 + dz^2 $$
and the metric for an expanding universe in comoving coordinates is:
$$ ds^2 = -c^2dt^2 + a^2(t)(dx^2 + dy^2 + dz^2) $$
So the only difference is that factor of $a(t)$, which we call the scale factor. This split between time and space is very similar to the split we use in flat spacetime, and it chooses a time coordinate we call comoving time. The nice feature of comoving time is that all comoving observers share this time coordinate i.e. they all agree on its value. As a rough guide, a comoving observer is one who is stationary with respect to the cosmic microwave background, and most objects in the universe are roughly stationary wrt the CMB. That means most observers in the universe are comoving, and therefore share the same time coordinate. That's why choosing to split spacetime into space and comoving time is so convenient.
So to return to your question, yes we are splitting spacetime into separate time and spatial bits. However this is a natural split that turns out to be very useful for describing the universe around us, so it is far from arbitrary.
