At what points do $\hat{ H}_{\rm high}$ and $\hat{H}_{\rm low}$ start to deviate from the ideal cases? Are there relatively simple models for these situations? (i.e., when does $\omega_0 << \omega_c$ become $\omega_0< \omega_c$?) Or when do the phases come into play?

Well, the question ``when do you deviate from the ideal case?'' is one that doesn't have a single answer; the answer is really ``it depends on how good you need the answer to be''. The exact filter response is usually possible to calculate, and you often can figure out the difference between the simple approximation and the more complete calculation. For example, if you eyeball the plots in the handouts, you can see the actual filter response with a smoothly curved transition between the $\omega<< \omega_c$ and $\omega >> \omega_c$ regimes. For some applications, treating the frequency response with straight lines might be perfectly good enough, even in the corner region. However, if you need to know with precision what your output waveforms will look like at frequencies near $\omega_c$, then you might want to do an analysis using the actual filter response with the curved line and the full $\hat{H}(j\omega)$ transfer function. (Of course, even the ``actual filter response'' with the curved line is itself an idealization, since it assumes resistors and capacitors are ideal circuit elements- but in real life all abstractions are leaky!)

Whether the phase shift matters or not depends on your application also. Relative phase shifts matter when combining signals.