About representation theory of Lie algebras: physicists actually care about that quite a bit, as the the theory of spin is intimately tied up with the subject. Your point stands, though. I don't know of any major applications outside quantum mechanics, and other parts of mathematics have even fewer applications. (I have always found integration theory kind of tedious for exactly that reason---some results turn out to be handy, but it's more like you're laying the groundwork for background material for stuff that'll be useful to physicists.) Also, the way in which physicists care about the math is very different from the way mathematicians do: we want the moral reasoning---we want to have some intuition for why the result is true---but (to grossly stereotype) we don't really care about the detailed proof.

(I must add that I heartily second your recommendation of /Q.E.D./)