Distribution of masses in GR creating Schwarzschild's solution? Suppose that in a region R of spacetime the Schwarzschild solution applies. Does it follow that the distribution of masses in the spacetime has spherical symmetry? Of course, these masses must be located outside of R since the Schwarzschild solution is a vacuum solution.
Note also that the converse to my question holds true: if the distribution of masses has spherical symmetry then the spacetime must be spherically symmetric, and then Birkhoff's theorem shows that Schwarzschild is the only vacuum solution.
 A: 
Does it follow that the distribution of masses in the spacetime has spherical symmetry?

No. It is quite possible to have a region (finite or infinite) with a Schwarzschild metric in it sourced by matter distribution without spherical symmetry.
To gain some intuition about the problem consider a similar question asked about electromagnetic field: “Can there be non-spherical distributions of charges that produce spherically symmetric (that is Coulomb) fields inside a region?” In answer, let us consider any finite static distribution of charges and enclose that distribution inside thin conducting sphere. The electric field inside the sphere would induce surface charge density on the sphere such that outside of the sphere the electric field is precisely Coulomb. All the necessary quantities could be calculated using the method of images.
This could be easily carried over to Newtonian gravitation with minor differences: (1) There is no simple mechanism that would “automatically”  generate the necessary surface densities, so those must be arranged “by hand”. (2) If electrostatic solution has negative surface charges, then it is possible to add an overall constant to surface density until all charges are positive so that solution directly translates to gravity.
Of course, once we go from electrostatics to Newtonian gravitation, the surface of the sphere is no longer the physical reason for a Coulomb field outside, but merely a device for simplifying the calculations. So we could, in fact construct monopolar gravitational field using bodies of arbitrary form and using either surface or volumetric mass density. This, in fact, was done in the paper:

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*Connes, Alain, Thibault Damour, and Pierre Fayet. (1997). Aspherical gravitational monopoles. Nuclear Physics B 490.1-2, 391-431, doi:10.1016/S0550-3213(97)00041-2, arXiv:gr-qc/9611051.

(See also question on Physics SE).
If we then try to go from Newtonian gravity to general relativity one would expect that aspherical sources of static monopole gravitational field would persist perturbatively in the weak field regime of GR. But one could also look for exact results. For example in the paper:

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*Bičák, J., Lynden-Bell, D., & Katz, J. (1993). Relativistic disks as sources of static vacuum spacetimes. Physical Review D, 47(10), 4334, doi:10.1103/PhysRevD.47.4334.

the authors showed how to construct spacetimes that have well known static vacuum axially symmetric metrics (including specifically the Schwarzschild metric) in two semi-infinite regions on either side of the infinite thin disk.  This disk which serves as the source of the gravitational field, has only axial but not the spherical symmetry.
Another interesting result by Carlotto & Schoen:

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*Carlotto, A., & Schoen, R. (2016). Localizing solutions of the Einstein constraint equations. Inventiones mathematicae, 205(3), 559-615, doi:10.1007/s00222-015-0642-4, arXiv:1407.4766.

demonstrates that it is possible to construct initial data for gravitational Cauchy problem that “confine” gravitational field for the matter carrying mass-energy to the interior of a spatial cone, while outside of the cone the initial data are strictly Minkowskian. This could be of interest to the OP's question (1) because Minkowski spacetime could be seen as a limit of Schwarzschild spacetime (2) the techniques of that paper allows one to construct spacetimes with multiple regions each with its own Schwarzschild field, so that those masses do not “feel” each other's gravitational field for a certain interval of time since initial moment.
Of course, when evolved, such initial data would give rise to a dynamical spacetime and the regions where this “shielding” occurs would be shrinking, as gravitational waves propagate across the spacetime.
