Characterizing the phase diagram for proteins is important both for laboratory studies and for the development of structure prediction algorithms. Using a variational scheme, we calculated the generic features of the protein thermostability over a large range of temperatures for a set of more than 50 different proteins using a model based on native structure alone. Focusing on a specific system, protein G, we further examined, using a more realistic model that includes the normative interaction, the thermostability of both the native state and a collection of trap structures. By surveying the native structures for many proteins and by paying closer attention to the various trap structures of protein G, we obtained an overall understanding of the folding dynamics far from the conditions usually focused on; namely, those near the folding temperature alone. Two characteristic temperatures (shown to scale with folding temperature in general) signal drastic changes in the folding mechanism. The variational calculations suggest that most proteins would, indeed, fold in a barrierless manner below a critical temperature analogous to a spinodal in crystallization. For fixed interaction strengths, this temperature, however, seems to be generally very low, similar to 50% of the equilibrium folding temperature. Likewise, native proteins, in general, would unfold in a completely barrierless way at a temperature 25% above folding temperature according to these variational calculations. We also studied the distribution of free energy profiles for escape from a set of trap structures generated by simulations.