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You have shared a lot of thoughts here. Let me briefly respond like this:
(1) When you ask, “Is there always an equilibrium?” it depends on the framework. For mathematical economists, they have “models” and in the model, there may or may not be equilibrium.
For Austrian economists, there are different categories of equilibria and one of them–the “plain state of rest”–comes to pass all the time. (I think Joe Salerno gets into that stuff in this article: 1994. “Ludwig von Mises’s Monetary Theory in the Light of Modern Monetary Thought.” Review of Austrian Economics 8: 71–115.)
(2) Regarding static/dynamic and electrical engineering: There are some economists who are turning to more sophisticated computer simulations. So rather than build a “tractable” model that can be solved analytically, they just populate a computer simulation with agents following certain rules (like “maximize utility” or “maximize profit”).
There are economists trying to make analogies with physics but I am skeptical of their approach. I think you throw out a lot of knowledge when you treat people as mindless.