Question 1. How can one distinguish the sphere from a plane (Flatland) based solely on experiences on the surface?

Question 2. How can one distinguish the sphere from a torus based solely on experiences on the surface?

Use shadows?
    Look at shadows at the same time of day? This is a "local"  feature of the surface.

Observe "curvature"?
    This is also a local property.

Circumnavigate ("global")?: 
    Go West -> return from the East, then  go North-> return from the south.
    On a sphere: There will always be 2 points of intersection of the curves determined by the two routes.
    On a torus: There will be only one point of intersection of the two routes.
    On a plane: Go West ->  you keep going... there seems to be no return???