-The Sylow theorems are a great example. I think they are taught so ubiquitously because they provide perhaps the most elementary example of the broad pattern in mathematics of determining the structure an interesting object (finite groups) according to an invariant (the order). They also illustrate the power of letting groups act on things. Still, I have never encountered an actual use of the Sylow theorems in real life.

-Jordan / Rational canonical forms. This is a fairly ubiquitous topic in undergraduate / 1st year graduate linear algebra, but it doesn't seem to come up all that often (I can think of a few applications in Lie theory and differential equations). And even when it does come up, you usually just need that the diagonal + nilpotent decomposition exists - you never need to actually do calculations.

-Riemann integration. Many undergraduates take a course based on, for example. Spivak's "Calculus" and/or Rudin's "Principles of Mathematical Analysis" in which Riemann sums are developed in great detail. I went through all that and I barely remember how it works, because now I have my good friend the Lebesgue integral. Of course the Lebesgue integral is more sophisticated and difficult to learn, while undergraduates should have SOME theory of integration available.

-Basic number theory. I remember computing stuff like 7^92 mod 11 and solving x^65 = -1 mod 5, but for the life of me I don't remember half of that stuff. I guess that sort of thing provides a good invitation to more difficult mathematics, and it is probably foundational for actual number theorists.

-I wouldn't really put point set topology on the list. As you say many logicians an algebraic geometers (even number theorists) encounter certain ideas in point set topology, and such notions are also very important in a different way to analysts (e.g. weak topologies, Frechet topologies).

2)

-Homological algebra. This is probably changing these days, but many students aren't seeing basic homological algebra until their first year graduate algebraic topology course.

-Representation theory. There are plenty of very nice, basic theorems that for some reason don't make it into the undergraduate or even graduate curriculum. I'm into analysis and geometry, and even for me I found this to be a big gap in my education.

-Lie theory. I learned it as an undergraduate, but I think most students don't.

-Metric geometry / convex geometry. There are lots of useful ideas here for analysts and even number theorists that aren't commonly taught.