A is A is not "metaphysical" it is epistemological. Your father is your father, no matter how much he changes. If this is not true, you could not even state your criticism. What you mean by, "same wooden chair?" does that have a meaning? If it does not, your quesion is meaningless. If it does, apply it to the present case.
If we want to actually use "A is A" for anything useful...
Are algebra and symbolic logic a problem for you. I'll humor, your father is your father.
Plurality. (There must be more than one thing.)
Your criticism begins, This axiom requires us ... and falls down there. Remember, you cannot use or imply the concept you are trying to deny in its refutation. You've already implied a plurality.
Dynamism. (There must be change.)
Your criticism, This contradicts "A is A," (your father never did anything?) and it also requires you to assume ... an assumption (or any other mental phenomenon is an action and all action requires change)You cannot imply an action in your refutation of dynamism.
Differentiation. (All existents are different.),p> Your criticism, A mighty broad statement, the truth of which is muddied by such things as quantum entanglement, not to mention Dr. Heisenberg's little theorem.
You've made way too much of this, and your criticism is totally irrelevant. (It might not be irrelevant to what you think it says.) It only means if there are two things which are really two things and not aspects of the same thing, for example, there must be something different about them, or they will not be two things. There must be some quality that is different for any two things, even if every other quality is the same and the only difference is their location, for example.
Hank
If there is dynamism, then it means at least some things can change. "A is A," implies that nothing can change. So either dynamism is not universally true, or "A is not always A."