0. It's only 10 equations if you don't express it in tensor form. If you express it as a tensor, it's one equation. But that's "just" math, not physics. It's a lot of math, but worth learning.
1. All the equations must be satisfied simultaneously to get a satisfactory solution. But that solution is limited to "classical" gravity---it doesn't deal with the other forces such as electromagnetism and the nuclear force. The Theory of Relativity is only a theory of gravity.
2. Today's Cosmologists say that the geometry of the Universe is close to flat within our event horizon. But that doesn't make it Euclidean, except as an approximation on certain scales and in certain dimensional directions. Being "flat" or nearly flat is one of the many things that a non-Euclidean space can be. (In other words, Euclidean geometry is included in non-Euclidean geometry as one of an infinite number of possibilities.) We can clearly see, for example, the non-Euclidean nature of the space near the Sun, in the orbit of Mercury and the trajectory of starlight.
I point out, however, that thinking of solutions of the Einstein field equations as non-Eulidean geometry is only an interpretation, or model, of the solution to these equations. In General Relativity, space itself is not "real", only spacetime events are real. Spacetime events appear to be taking place in and connected by a geometry which is most easily (and beautifully) interpreted as non-Euclidean. But the space itself is "nothing" and has no measurable properties. Space is not itself a thing, and its geometry is a mathematical convenience with no observable consequences aside from the relation between actual spactime events. This is not the case with quantum field theories, where the space is full of fields such as the electromagnetic field. Reconciling these two different ideas about the "vacuum" is one of the tasks of a "Theory of Everything".
