Do Einstein field equations only relate local spacetime curvature to local energy-momentum of matter? Do Einstein field equations only relate local spacetime curvature to local energy-momentum of matter?
If so, can we extend Einstein field equations globally relating global spacetime curvature to global energy-momentum of matter?
 A: Yes, they cannot be extended to relate global curvature to global energy-momentum, not in general at least. You can see this by noting that the Einstein Field Equations can be derived by demanding that the stress energy tensor is locally conserved:
$${{\nabla}^{μ}T_{μν}} = 0$$
and noting that
$$\nabla^μ(R_{μν}-\frac{1}{2}g_{μν}R+Λg_{μν})= 0$$
is an identity in (pseudo) Riemannian geometry.
In general, the Einstein Field Equations cannot be extended to relate global spacetime curvature to global energy- momentum because global energy-momentum is not always well-defined. Noether’s theorem relates conservation of energy to time translation symmetry. In spacetimes like the friedmann expanding universe for example, time translation symmetry is not obeyed so no such globally conserved quantity can be defined.
In specific spacetimes you may be able to define a globally conserved energy. However the fundamental idea of general relativity is local, not global.
A: In the limit for small velocities and masses the Einstein solutions must match the Newtonian ones.
Suppose a point inside a tunnel from pole to pole of a spherical planet without atmosphere. The point is vacuum locally because a small neighborhood of the point is in the vacuum, but the Poisson equation $\nabla^2 \phi = 4\pi G \rho$, where $\rho$ is the density of the planet, is valid, which can be verified by substituting the known solution $$\mathbf a = -\nabla \phi = -GM\mathbf r$$ Where $M$ is the mass of the planet.
The same must be valid for the Einstein field equations. In this case, $T^{00}$ is related to the density of the planet as a whole, and not locally to a small neighborhood of the point (which would be zero). The Ricci tensor is also not zero as in vacuum solutions.
