Confusion regarding temperature and entropy

If the temperature is non-uniform, then the system is not at equilibrium and none of this applies.

The entropy (from the perspective of statistical mechanics - the number of microstates) is always well-defined (since is fundamentally combinatoric and does not depend on a given configuration of the system), but the temperature you obtain by this process is the temperature of the system, with a given energy $E$, when it finally reaches equilibrium.

You can eventually do this locally, if the system takes a long time to reach equilibrium, but it will only be an approximation.

Although the subject is called thermodynamics, it is more like thermostatics, since there is no dynamic (there is no time). The study of non-equilibrium thermodynamics (which is actually thermodynamics) and the definition of quantities, like the temperature, in these systems is a field of research.

Actually, you can easily notice that you need to be working in the context of equilibrium. The laws of thermodynamics imply that a system with non-uniform temperature eventually thermalize. This would introduce some notion of time, as you would need to label your quantities. The equilibrium properties are the only ones that do not depend on time.

Some people feel that it is valid to evaluate thermodynamic state properties locally (i.e., per unit mass), and I am one of them. In the case of temperature, for example, the partial derivative of specific internal energy with respect to specific entropy at constant specific volume would be the local temperature. Without assuming such local validity, it would be impossible to get accurate solutions to transient systems and steady flow systems experiencing transport processes. So we could never design heat exchangers, cooling towers, boilers, chemical reactors, distillation columns, absorption columns, piping systems, or any other industrial scale chemical processing equipment.