My understanding of General Relativity
Most of what you said is right.
If the energy-momentum tensor is known, the Einstein field equations can be used to solve for the metric tensor
This is wrong. For example, suppose the energy-momentum tensor is zero. There are still many possible metrics, including Minkowski space, versions of Minkowski space with nonstandard topologies, spacetimes containing gravitational waves, and black-hole spacetimes.
the Schwarzschild metric is the solution for the metric tensor if the energy-momentum tensor is that of a spherical star or black hole
The energy-momentum tensor of the Schwarzschild metric is zero everywhere. The mass of the black hole is hard to pin down. You can think of it as being at the singularity, but the singularity is a spacelike surface in the future and is not part of the spacetime manifold. Or you can think of the mass as being in the spacetime but not being localized, but then it's not measured by the energy-momentum tensor.
Does the energy-momentum tensor vary with the spacetime coordinates just like the metric tensor does and is it determined by the distribution of energy and momentum throughout spacetime (i.e. if a massive body exists somewhere)?
The energy-momentum tensor does vary from point to point. Its value at a point only describes the energy and momentum density at that point, not far away.
If that is so, does the value of the energy-momentum tensor at a point in spacetime influence the curvature of spacetime only at that specific point or does it influence the curvature of surrounding points in spacetime as well (i.e. does the Sun cause spacetime to curve in a large region around it or just at the points in spacetime where the Sun exists)?
This depends on what you mean by "influence" and "curvature." There is curvature that is not measured by the Einstein tensor, such as the curvature of a gravitational wave. The direct influence of the stress-energy is only on the part of the local curvature measured by the Einstein tensor.
This is actually pretty similar to electromagnetism. The divergence of the electric field is determined locally by the charge density, but electric fields propagate.
It's great that you're formulating questions of this kind. These are all good, natural questions to be asking as a beginner at GR. Good luck!
If the energy-momentum tensor is known, the Einstein field equations can be used to solve for the metric tensor (i.e. the Schwarzschild metric is the solution for the metric tensor if the energy-momentum tensor is that of a spherical star or black hole).
The metric tensor depends also on the symmetries. For example: if the energy-momentum tensor is zero in a region outside a spherical mass, and this mass is not rotating, we can say that there is a spherical symmetry, and the field is only a function of $R$. After calculating all components of the Ricci tensor, we come to differential equations that leads to the Schwartzschild metric.
In this approach we don't use any information about mass or energy values or densities. It is forcing the equation to match Newtonian gravity for weak fields that brings the product $GM$ to the metric.
But if this mass is rotating, the spherical symmetry is no more valid and the metric is different.
First, you need boundary conditions as well as the energy momentum tensor to determine a solution to Einstein's equation for gravity.
In answer to Q1, tensors, including the energy-momentum tensor and the metric tensor, are coordinate independent. In practice, calculation requires a choice of coordinates. It is the representation of the tensor in given coordinates which varies, not the tensor itself.
In answer to Q2, Einstein's equation
$$ G^{ab} = 8\pi G T^{ab} + \Lambda g^{ab}$$
states that Einstein curvature $G^{ab}$ is specified at a point by the energy momentum tensor (and cosmological constant). It does not specify the Riemann curvature tensor $R^a_{bcd}$. The Riemann curvature tensor can be found from the solution of Einstein's equation given $T^{ab}$ together with boundary conditions. IOW the Sun does indeed cause spacetime to curve in a large region around it.