How would you define the sorts of tasks humans are designed to carry
out? I can't see an easy way of categorising all the problems
individual humans have shown there worth at, such as key-hole surgery,
fighter piloting, cryptography and quantum physics.
Well, there are two timescales involved, that of the species and that of the individual. The short answer to the first question is: survival in Stone Age tribes on the plains of Africa. That this produced an entity that can do all the things on your list invokes something between wonder and existential paranoia depending on one's mood and predilections. (The absence of any steps of the Great Filter between the Tertiary and the Cold War is a common assumption - but it is only an assumption. But I digress.)
On the individual timescale we're programmable general purpose problem solvers: We're good at learning from our environment, but that only gets you so far, by itself it won't let you do any of the above things because you'll be dead before you get the hang of them. However, our environment also contains other people and we can do any of the above by learning the solutions other people worked out.
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