2 added 73 characters in body

The "only if" direction in 3 is incorrect: take standard $\mathbb R^2$ and introduce several bumps to make the isometry group trivial. The "if" direction is the Kobayashi-Nomidzu "Foundations of Differential Geometry", vol I, around Theorem 4.6 for references. There and in vol II you also find answers to both of your 1-2 questions, I think.

To see how sectional curvature is computed you need to go through a lot of examples.

I think the

The "if" direction in 4 is incorrect: there are manifolds of constant scalar curvature that are not locally symmetric.

1

The "only if" direction in 3 is incorrect: take standard $\mathbb R^2$ and introduce several bumps to make the isometry group trivial. The "if" direction is the Kobayashi-Nomidzu "Foundations of Differential Geometry", vol I, around Theorem 4.6 for references. There and in vol II you also find answers to both of your 1-2 questions, I think.

To see how sectional curvature is computed you need to go through a lot of examples.

I think the "if" direction in 4 is incorrect.