I have two favourite books on differential geometry where you can find answers to your questions:
- do Carmo's Riemannian Geometry (as suggested by David Lehavi)
- 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.