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I have two favourite books on differential geometry where you can find answers to your questions:

  1. do Carmo's Riemannian Geometry (as suggested by David Lehavi)
  2. Besse's Einstein manifolds

Let me just point out that your 4th point is not quite correct. The statement is that

A riemannian manifold is locally symmetric if and only if the Riemann curvature tensor is parallel with respect to the Levi-Civita connection; i.e., $\nabla R = 0$

This presupposes that by "locally symmetric" you understand that the geodesic symmetry (i.e., changing the sign of the parameter of the geodesic) is an isometry at every point.