As short as possible:
From what I understand the Cosmic Microwave Background was predicted by multiple people as a theoretical consequence of the Big Bang theory, and hence today, is evidence for the Big Bang. So of course there was also a prediction of the age of the CMB, because it was predicted to have occurred at a certain point in time.
Now, is that the only reason cosmology dates the CMB to 13.8 billion years ago, i.e. is the dating inferred by the theory? I assume there must be many types of measurements that somehow connect to this so as to infer the exact age of what we are seeing there.
Best Answer
You have asked a lot of questions here so my apologies if I do not cover them all. Let us start with a misconception which you may (or may not) hold:
The CMB does not originate from the big bang it originates from what is called the surface of last scattering. This occurred about $380,000$ years after the big bang when the universe had a temperature of about $0.23-0.25eV$ ($\sim 3000K$). Consider the reaction: $$p+e^-\leftrightarrow H+\gamma \tag{(1)}$$ above $3000K$ the photons have enough energy to cause this reaction to go into reverse once the universe as cooled (due to the expansion of the universe) the energy of the photons is to small for the reverse reaction to occur. The consequence of this is that all the protons and electrons get taken up to form hydrogen atoms. With no free ions in the universe Thomson scattering becomes less probable ans the universe becomes optically thin. I.e. Before this time photons would travel a very small distance before scattering but after they will travel a very large distance - so large that the photons we see as the CMB are the photons from this time which are yet to scatter.
Now this surface of last scattering will occur at $3000K$ no matter what happened before it (this is determined by something known as the Saha equation). So only the rate of expansion after this time will effect the redshift of the CMB. Now inflation is thought to have happened within approximately $10^{-32}$ seconds after the big bang. Clearly $10^{-32}$ seconds $\lt $ $380,000$ years so inflation will not affect the redshift of the CMB.
Now you actually pose another query in your question - whether the cosmic horizon will change during inflation. The answer to this is yes. And no there is no contradiction here - the cosmic horizon is different from the surface of last scattering. The Cosmic Horizon is the furtherest point we could possible see. I.e. if a photon started traveling at the big bang singularity and has carried on in a straight line (without scattering) to reach us today - the distance it has traveled is the cosmic horizon. This is affected by inflation because the photon is assumed to be traveling through this period.
All that said inflation does have an effect on the CMB. It explains why it looks so homogeneous (something called the horizon problem). Inflation explains this since it means a very large are of the universe now was once all in causal contact since before inflation is was much smaller - allowing photons from one side to the other.
EDIT
Concerning the question of evidence of the age of the CMB. We can only measure the CMB as it appears to us now. This includes things like the wavelength, blackbody spectrum and the angular power spectrum of the anisotropies of the CMB. To work out the age of the CMB however, you need to know the temperature of the CMB when it was created and compare that to its value today (which we can measure). There is no way to measure the temperature when it was created (we simply weren't there) and thus for this we need the theoretical calculation involving the Saha equation.