You can think of the scene you see in a reflection as being for all optical purposes like a 3D object located behind the mirror: the image. This is because rays of light emanating from the real object, no matter what direction they are going in, reflect off the mirror at such an angle that they appear to have come from the direction and distance of the object's image and therefore behave in exactly the same way as if they really did come from an object located at the image.
(A not all that helpful explanation and diagram is at
Virtual image - Wikipedia, the free encyclopedia )
For a plane mirror, distance from the image to the mirror equals the distance from the object to the mirror. That's when the rule you quoted about the sum of the distances would apply.
For a convex mirror, the image is closer (but smaller than the object - see pic on wiki page) so the focal distance will need to be shorter than the sum and the DOF will be that you expect of a (smaller) object at that (lesser) distance.
For concave, it's potentially vice versa (but it gets more complicated here with inverted images etc - see
Curved mirror - Wikipedia, the free encyclopedia ).
If you really wanted to do the calculations to work out your image distance and DOF (I wouldn't... talk about spoiling the fun of photography
), it can be done pretty easily. Let f = focal distance = half of the radius of curvature of a spherical mirror. Then the lens equation, 1/f = 1/u + 1/v will allow you to take the distance (u) from any point on your object and find out the distance (v) between the mirror and the corresponding point of the image, or vice versa. (v will be negative if the image is behind the mirror)