Could a Dyson Sphere-like object be made to encase a black hole without collapsing? In this question, I'm not asking about the feasibility of building a black hole Dyson Sphere or the effectiveness, I simply want to know if an object could stay in a locked position encasing a black hole without collapsing in on itself. It doesn't matter if it would collapse before in place, I just care about the final position and if so, what shape would be best suited for this?
 A: 
… if an object could stay in a locked position encasing a black hole without collapsing in on itself.

Yes.
Solutions for a thin static spherical shells around a nonrotaing black hole have been considered in the following paper:


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*Frauendiener, J., Hoenselaers, C., & Konrad, W. (1990). A shell around a black hole, Classical and Quantum Gravity, 7(4), 585, doi:10.1088/0264-9381/7/4/011, free pdf.


The authors found that for a reasonable constraints on the equation of state for the shell material it is possible to have a shell around the black hole, even possibly inside the radius of circular photon orbit, but necessarily bounded away from the horizon.
Another paper of note:


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*Brady, P. R., Louko, J., & Poisson, E. (1991). Stability of a shell around a black hole. Physical Review D, 44(6), 1891, doi:10.1103/PhysRevD.44.1891.


These authors have considered mechanical stability of such a shell and found that …

… the shell is stable against radial perturbations only outside a critical radius which is always larger than the radius of the circular photon orbit. The surface energy density of a stable shell is always larger than twice the surface pressure, and thus satisfies the dominant energy condition by a wide margin.

So, a mechanicaly stable Dyson sphere must be located some distance away from circular photon orbit, but it is conceivable that some active control system could stabilize Dyson spheres even inside this radius.
