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Here is a partial answer to the second question. Among other things in Trnková, Věra Topological spaces with prescribed nonconstant continuous mappings. Trans. Amer. Math. Soc. 261 (1980), no. 2, 463–482 it is shown that given any monoid M, there is a topological space X such that the non-constant mappings on X form a monoid isomorphic to M.