Are there any aircraft designed to operate between an altitude of 50-80km?
Is there any aircraft designed to perform its mission into this 30km wide slice of atmosphere? 


(between 50km and 80km) - apart from some experimental rocket planes and so on, I can't think of any. The main problem is there's so little air - very little opportunity to generate lift, and very little air to run a jet engine.

In theory it should be possible for a winged aircraft to fly up there (it is below the Kármán line) but the engineering difficulties would seem to make this impractical.  For more detail I'd recommend the top answer for this question: What determines the maximum altitude a plane can reach?


  What is the usefulness of this slice for aviation?


If it were possible, it would be useful for futuristic or military things like:


spy planes like the SR-71 but higher and faster
super-fast suborbital transport
passing through on getting into orbit - single stage to orbit
vehicles for example.


(All of these examples are pipe dreams at the moment of course.)