Solve mirror equation for image distance.
A spherical mirror's focal length is half its radius of curvature, positive for a concave mirror and negative for a convex one. With that sign in place the mirror equation has the same reciprocal form as the lens equation, so the image distance follows directly. The sign convention does the interpretive work: a positive image distance means real rays converge in front of the mirror, while a negative one means the image only exists as a projection behind it.
Spherical mirror equation
f = R/2 (negative for convex); 1/f = 1/do + 1/di, so di = 1 / (1/f - 1/do); m = -di/do
Because it diverges reflected rays rather than converging them, so the focus lies behind the surface. That is what makes convex mirrors produce a permanently virtual, reduced, upright image with a wide field of view — the reason they are used as vehicle wing mirrors.
Inside the focal length. Then the image distance comes out negative, giving a virtual, upright, enlarged image — the shaving or makeup mirror case. Beyond the focal length the image is real and inverted.
Yes, it is the paraxial approximation. Rays far from the axis on a genuinely spherical surface focus at slightly different points, an aberration real telescopes avoid by using a parabolic surface instead.