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One idea that comes to mind: Geoff Cruttwell and some of his students have had some fun counting the number of categories with some fixed finite number of morphisms, which is in some sense a fairly elementary problem. I expect there are related questions that are still open; you could contact him and ask if he has ideas. For instance, one thing you could do with finite categories is compute their Euler characteristics, and then perhaps do statistical analyses of these too.
Monoids dominate of course. And of those which aren’t, it looks like disjoint sums of monoids dominate. How about counting minimal representatives of equivalence classes of categories which require at least two objects with no disjoint subcategory?
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