However, plane properties do not hold on a larger scale. Consider, for example, a triangle on the surface of a sphere. It doesn't need to have 180 degrees for the sum of its interior angles. Consider a triangle starting at the pole, go down to the equator, travel the equator for a 1/4 of a circle, then go back to the pole. The sum of the interior angles in this "triangle" is 270 degrees. In a similar manner many other Euclidean properties of figures do not hold.
Check this with the other (more qualified) members of the forum, but I would imagine the best definition of the angle made by two intersecting lines on your surface would be to "zoom in" on the point of intersection to look at things locally, and use the Euclidean definition of an angle. Needless to say, the same angle at two different points on your suface may wind up looking vastly different on a macroscopic scale, so the geometry could include some unusual properties compared to Euclidean geometry.
I've never heard of this and it boggles my vastly limited visualization abilities. Where is the second center??Also, every great circle has TWO centers.