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"The Einstein's field equation is a tensor equation relating a set of symmetric 4 x 4 tensors. It is written here using the abstract index notation. Each tensor has 10 independent components. Given the freedom of choice of the four spacetime coordinates, the independent equations reduce to 6 in number."

So is it true to say that the Einstein tensor is a symmetric 4 x 4 tensor which consists of 10 independent highly non-linear equations. Does this imply we are talking about a general 4 dimensions of space-time? But what I don't understand is the last bit. It says given the freedom of choice of the 4 space-time coordinates, the number of independent equations are now down to 6. Could someone explain this bit to me please? I just can't see how it works.

Finally, assuming that the cosmological constant in Einstein's equation is not zero, could one still replace the metric tensor by the flat metric to solve Einstein's equation?

Thanks