As I have remarked in the comments, it is not clear whether the questions refer to particular statements made in the linked article or to the Dicke model in general. I therefore address them in the latter sense:
- Steady state solutions are different on different sides of transition. In non-linear theory (which could be also used here) this is referred to as bifurcation, which may be characterzied by (dis)appearance of some solutions, change of their number, or change of their stability. Thus, in the classical presentation of the Landau theory of the second order phase transitions one passes from a potential with one extremum (minimum), which corresponds to a unique solution, to a potential with three extrema (two minima and one maximum), which describes two stable and one unstable solution. Similarly, the first order phase transition in Landau theory corresponds to one solution losing its stability, whereas the other becomes stable.
- The removal of the counter-rotating terms (rotating wave approximation) has to do with the scales of coupling and frequency (see, e.g., this question and the answer). When the two are of the same order, the rotative wave approximation is simply not justified. On the other hand, the phase transition in the Dicke happens is easiest to achieve when the frequencies and the coupling strength are comparable, since the critical point is given by $\lambda_c=\sqrt{\omega\omega_0}/2$.
