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The second point first: The only reversible processes--in principle--are isentropic. And they don't really exist in nature. One can only approximate them. The world would not evolve in any interesting way without entropy. The link you have posted is to Shannon's construction of entropy that wasn't conceived of until about mid 20th century. The concept of entropy, as you know, predates this mathematical construction by many, many years, and covers much more than information theory.
As to the e^ipi=-1. I prefer it that way, because that is its reality. The expression e^it loses its meaning (its rotation in phase space) if a constant is arbitrarily is added to it. I think its more beautiful to let it evolve with time and notice that it falls on -1 at precisely pi.