Why did CMB decouple when neutral atoms formed? Atoms can absorb and emit photons. But it is said that when the neutral atoms were first formed the photons got decoupled. If the atom-photon interaction is allowed how can we say that photons were decoupled? After recombination, does the interaction of neutral atoms with CMB photons (through absorption and emission) stop? Why? Thanks!
 A: During recombination the electrons enter higher energy states of system and then fall into the ground state.  There is then subsequent photon production and many of those photons were reabsorbed by the newly formed neutral atoms.
Consider photons scattering off of free electrons. You can calculate the rate of Compton scattering and compare this to the rate of the expansion of the Universe. At some point the rate of Compton scattering would be much lower as electrons become bound, and the mean free paths of the photons would be around the size of the horizon. At this point, they can be thought to be decoupled since the mean free path is the size of the horizon.
A: No, the interaction of atoms and photons does not stop, but it becomes small enough to be neglected.
At a basic level, the mean free path of a photon is $(n \sigma)^{-1}$, where $n$ is the number density (per unit volume) of the particles doing the absorbing/scattering of photons and $\sigma$ is the cross-sectional area of their interaction with the photons.
Before decoupling, the cross-section is essentially the Thomson scattering cross-section of free electrons - $\sigma = 6.6 \times 10^{-29}$ m$^2$ and is wavelelength-independent. The number density of electrons is about $1.6 (1+z)^3$ m$^{-3}$ and recombination occurs at around $z \simeq 1100$. Thus the mean free path for light (of any wavelength) is around 700 light years. But since the universe is around 400,000 years old at this point, this means that the universe is effectively opaque to the radiation that is within it (i.e. a photon could only travel a small fraction of the size of the observable universe before interacting).
After recombination the value of $\sigma$ drop precipitously, by many orders (at least three) of magnitude at almost all wavelengths (the exception would be the main discrete transitions in a neutral hydrogen atom). The density of hydrogen atoms is roughly the same as that of electrons before recombination, but because the cross-section is so much smaller, the universe becomes effectively transparent to the photons (at most wavelengths), and becomes more so as the universe expands and the density gets smaller.
