What I don't quite understand is how do we relate the parameters of Rabi cycles like Rabi frequency to the intensity of absorption? Why do we in fact see any absorption at all if it's more or less compensated by emission?
Rabi oscillation is a concept based on an approximate description of a single atom. This description does not include any irreversible element so it cannot by itself explain systematic absorption in medium.
If the model is to manifest behaviour consistent with systematic absorption of light, it has to include some form of friction - mechanism which allows the charge oscillations of the atoms to be not in phase with the external electric field.
One way to do this is to include mutual interaction of atoms in the medium into the description. The friction (and absorption) is then macroscopically visible result of the reversible interaction between charged particles.
Simpler but less accurate would be to just add some damping term into the Schroedinger (or equivalent) equation to make the oscillations of the atoms' electric moment lag in phase behind the external electric field.
Often all this is neglected and the absorption coefficient is obtained by a formal trick - assuming the atoms jump between discrete states irreversibly with probability given by the Fermi golden rule formula (see time-dependent perturbation theory). Then the absorption coefficient is calculated to be proportional to square of the transition dipole moment of the pair of eigenfunctions whose frequency of oscillation matches the external frequency. The absorption coefficient does not depend on the amplitude of the external electric field though, so it is not simply a function of the Rabi frequency.