Yes, several aspects of general relativity have been experimentally verified such as frame dragging from rotating gravitational sources and the recently discovered gravitational waves.
The Schwarzschild solution is a vacuum solution. There is no stress-energy tensor defined anywhere in the solution. That is what makes the Schwarzschild solution so simple. Schwarzschild black holes likely don’t physically exist since they are modeled as eternal, static solutions with no angular momentum.
The equations of general relativity are impossible to solve in the general case. The known exact solutions are all simplifications. What makes difficult solutions even more difficult is trying to model a matter/energy distribution where the stress-energy tensor is zero at some points and non-zero at other points. You essentially have a vacuum solution and a non-vacuum solution that have to be smoothly blended.
As for the stress-energy tensor itself, the Levi-Civita connection imposes a requirement for a rank 2 symmetric tensor as the source of gravity. It is not a big stretch to extend the energy-momentum 4-vector from special relativity to a stress energy tensor.
Now, are there other possible sources other than a symmetric stress-energy tensor? The answer to that is yes. If we modify General Relativity to allow a non-symmetric connection (introduces a spacetime torsion), we can then have a non-symmetric stress-energy tensor. The Einstein-Cartan modification does that. It postulates a torsion tensor that models quantum spins. Difficult to test because only a non-vacuum torsion will behave differently than vanilla relativity.
As for electromagnetism, yes there is a stress-energy tensor for electromagnetism. Testing would be difficult since the energy-momentum density of electromagnetic fields is so small.
Anything that possess energy-momentum in any form will gravitate.