functional analysis in probability theory, Feller processes, contraction semigroups

To your last point: see the wondeful paper by Nelson on the Feynman path integral. He uses semigroup techniques to make the formalism rigorous.

As a side, unitary groups play a fundamental role in quantum mechanics because of the canonical commutation relations.

For more applications of semigroup theory apart from probability, see the outline in Chapter VI of Engel and Nagel.