Does it make sense to measure the time from the Big Bang until the CMB was emitted? It is very common to hear cosmologists talking about what happened some time after the Big Bang.
Here is a good example of chronology:  https://en.wikipedia.org/wiki/Chronology_of_the_universe
The most common thing is that they say that the universe was sparse and cold enough 380.000 years after the Big Bang for light to travel which we see as the cosmic microwave background.
But I imagine that at that time (and even more up to that time) the universe was incredibly dense, heavy and energy-rich. In general relativity this affects the time, but how much? Does it make sense to count those 380.000 years.
 A: The answer comes in two parts.
First, when in cosmology people use the phrase 'since the big bang' they normally mean 'since some very early moment such as the Planck era'. One cannot trace the time right back to singular conditions, which would not be well-defined, but it is not necessary to do that. Just say 'we'll measure from some early moment when we have a well-defined notion of time, after the density and temperature have settled enough so that quantum gravity is not needed.'
Secondly, the time variable being used here is the one appearing in the field equations of GR, usually in a suitable approximate version such as Friedman equations. That means it is proper time at any given spatial location and thus it also corresponds to the time which appears in particle physics calculations at any given spatial location.
A: The following is aimed specifically at answering the two question you asked.
"In general relativity this [the universe was incredibly dense, heavy and energy-rich] affects the time, but how much?"
"Does it make sense to count those 380.000 years."
If you want to calculate the physical parameters (say for example: temperature or scale factor or the fraction of the atoms that are gas) at a much later age, say a billion years or more, then ignoring the 380,000 years won't effect your results very much. If you want to calculate a value (say for example: temperature or scale factor or the fraction of the atoms that are gas) during various phases of the transition from all atoms being a plasma to all being a gas, then a reasonably close approximation of the corresponding age (which will be somewhat near 380,000 years) is critically important.
A: Normally, if massive particles are present we can talk about the time scale set by their mass $t_C = \lambda_C /c$ where $\lambda_C = h/mc$ is the Compton wavelength.
However, their was an era during which the electroweak phase transition hasn't happened yet. During that era the expectation value of the Higgs field was 0 and hence all particles were massless. In this case, there is no universal time scale, it's undefinable. Physics during this era was simply scale invariant: $t \to at$ would result in the same. Hence, before the phase transition it's actually impossible to properly measure time.
Even then, the FLRW metric is defined using co-moving coordinates, but every particle moves at $c$ in every frame. We can't just move along with a particle because we can't find a rest frame for that particle. Moreover, as they are massless they all move on null geodesics: $ds^2=0=c^2 d\tau^2$ where $\tau$ is the proper time. From this we can see that $d\tau = 0$, so which proper time of which particle can we promote to cosmic time? None of them!
Once could construct a cloud of particles and use its centre of mass as co-moving particles, but in a scale invariant theory it's useless to measure distances. So this method also fails.
Talking about time so close to the Big Bang is highly nontrivial.
