The surface of a sphere is 2D so a position on it can be represented by two coordinates such as longitude and latitude. You have differant projections of the earth. Just using longitude and latitude as (x,y) coordinates is the common map. Directions are accurate, but distances are not. Over a small enough patch far enough from the poles distance can be measured accurately enough.

Correct distance calculation, or really more correct, is the great circle distance. The Earth is, more or less, a spheroid. It's sort of squished from spinning. With a sphere it's simplier, it's just the arc of a circle. You can find website that give all kinds of formulas for geometry such as Wolfram's MathWord site.

Matrix for a transform? From what? The implicit equation for a sphere is x^2+y^2+z^=r^2, not real useful. You generally construct a sphere using the parametric equation for a sphere, i.e. [x,y,z] = f(u,v) = [r*cos(u)*sin(v), r*sin(u)*sin(v), r*cos(v)] which is the formula for converting from 3D polar coordinates to rectalinear coordinates.