As a rule of thumb, you should get the point where you can teach yourself topics on your list without a formal class. That usually involves some amount of formal classes to get there.
How these are applied is not the same from job to job. Derivatives pricing is the focus of many MFE programs, I think in part because it involves different types of math (and in part just because of history). A derivatives pricer might have a model that involves a SPDE (so they need stochastic calculus to interpret the basic model), use statistics like regression to parametrize their model and use numerical methods in the likely case the SPDE does not have a closed form solution.
That's my understanding, but I am not working as a derivatives pricer currently.
In other branches, portfolio and risk functions involve using a lot of statistics and also understanding the measures of derivatives (Greeks) so that some knowledge of stochastics is also present but not to the extent of those working in derivatives.
In any case, you would encounter numerical methods and complexity theory (the theory of algorithms) just to make your codes work and do what they're supposed to do in a reasonable amount of time.
Things like measure theory, real analysis and functional analysis are more helpful in the sense that that background means you understand math pretty well and can teach yourself whatever. It more comes up in the sense of understanding journal articles when you are attempting relatively untried things than during implementation, at least in my experience.
