Dear Brian,
Thanks for your excellent essay. Let me comment only on zero-temperature dynamics. I limit myself to the initial condition with zero magnetization, or more generally magnetization of order N^{-1/2} where N is the total number of spins. This physically corresponds to the sudden quench from infinite temperature (more generally, from initial temperature above critical) to zero temperature. In addition to ground states, the system may fall into a state with an arbitrary number of stripes. There is an intriguing connection with critical percolation (an equilibrium problem!) that allows extracting exact probabilities for some outcomes, like the probability to reach a ground state. The answers depend on the boundary conditions and, more unexpectedly, on the aspect ratio (one goes to the thermodynamic limit keeping the aspect ratio fixed). More on that is in
https://arxiv.org/abs/0905.3521
https://arxiv.org/abs/1208.2944
The 2d Q-states Potts model (Q=2 corresponds to the Ising model) with zero-temperature Glauber dynamics is poorly understood. One sees much more rich behaviors, e.g. subsets of states changing ad infinitum
https://arxiv.org/abs/1305.1038
One generic lesson is that gradient descent (that is similar to the Glauber or Metropolis dynamics at zero temperature when only energy decreasing or conserving moves are allowed) can bring not to the paradise, the ground state with the smallest possible energy, and not even to a jammed state with locally minimal energy in some landscape of states, but to a connected set of equal-energy states where the system wonders ad infinitum.
