Angle sum of rays The sum of angles of a triangle depends on the curvature of the surface and can deviate from $\pi$. What about the sum of angles between successive lines emanating from a given point P? Can it deviate from $2\pi$, depending on the curvature at P? (I guess not.) How is it proved?

If $P$ is an ordinary point of a surface $S$ then the angles between "rays" emanating from $P$ are measured in the tangent plane at $P$ which is nothing but an ordinary euclidean plane with a distinguished point $O$. That the sum of the angles between rays emanating from $O$ is constant is a deep fact of elementary geometry, but this fact can be established without resorting to Riemannian geometry.

If $P$ is a "special" point of your surface $S$, e.g., the tip of a circular cone, where a positive amount of curvature is concentrated in one point, then the sum of the angles between rays emanating from $P$ is no longer $=2\pi$.