I think that the Einstein Field Equation relates the pseudo metric to the the distribution of matter-energy as represented by the stress-energy tensor. Are the stress entries in the stress-energy tensor restricted to the tidal forces on a (falling or orbiting) test blob, or do they include other mechanical stresses such as pressure in a stationary spherical gas cloud in equilibrium?
The stress-energy tensor is a stand-in for whatever forces you want to account for. For example, electromagnetic waves carry energy (as well as momentum, angular momentum, etc.) and you can choose to include these energies in the stress-energy tensor if you wish. Please see the Wikipedia article about the stress-energy tensor for E&M: https://en.wikipedia.org/wiki/Electromagnetic_stress%E2%80%93energy_tensor