Yeah, once you add in a second mass to a Schwarzschild spacetime you'll have a new spacetime that can't be written as a "sum" of two Schwarzschild spacetimes, depending on the specifics there could be ways to simplify it but I doubt by much.

If GR was linear, then yeah the sum of two solutions would be another solution just like it is in electromagnetism.

I'm actually not 100% certain how you'd treat a shell, but I don't think it'll necessarily follow the same geodesic as a point like test particle. You'll have tidal forces to deal with and my intuition tells me that will give a different result, though it could be a negligible difference depending on the scenario.

Most of my work in just GR was looking at null geodesics so I don't really have the experience to answer that question conclusively. All that said, from what I recall it's at least a fair approximation when the gravitational field is approximately uniform, like at some large distance from a star. The corrections to the precession of Mercury's orbit were calculated with Mercury treated as a point like particle iirc.

Close to a black hole, almost definitely not. That's a very curved spacetime and things are going to get difficult, even light can stop following null geodesics because the curvature can be too big compared to the wavelength.

Edit: One small point, the Schwarzschild solution only applies on the exterior of the spherical mass, internally it's going to be given by the interior Schwarzschild metric.