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JeremyHahn comments on Second-Order Logic: The Controversy - Less Wrong
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I do not think the situation is as simple as you claim it to be. Consider that a category is more general than a monoid, but there are many interesting theorems about categories.
As far as foundations for mathematical logic go, any one interested in such things should be made aware of the recent invention of univalent type theory. This can be seen as a logic which is inherently higher-order, but it also has many other desirable properties. See for instance this recent blog post: http://golem.ph.utexas.edu/category/2013/01/from_set_theory_to_type_theory.html#more
That univalent type theory is only a few years old is a sign we are not close to fully understanding what foundational logic is most convenient. For example, one might hope for a fully directed homotopy type theory, which I don't doubt will appear a few years down the line.
Sure. It has of course been the case that careful increases in generality have also led to increases in power (hence the weasel word "usually"). There is something like a "production-possibilities frontier" relating generality and power, and some concepts are near the frontier (and so one can't generalize them without losing some power) while some are not.