Actually it is simplier to begin from (2)
Explanation of the terms: first term is the vector parallel to e, second orthogonal to e
A x e is normal to both A and e and has magnitude of the projection of A on the plane normal to e, and e x ( A x e ) has the same magnitude and is normal to e and previous vector..
(1): express X as shown in (2)
(3): pay attention that the curve is a circle (try to make sure..).. as you probably know, the force applied on a mass rotating with a constant speed is normal to the direction of its movement (and so is acceleration), so the dot product is zero..
about derivations in polar coordinates: locally, at any point you have ortogonal axes r and theta, so locally there is no difference between cartesian and polar coordinates, and that's what you need to know for finding derivatives. making dot and cross products in spherical coordinates is not nice (but there are formulas for that), so it is better to go though cartesian coordinates.