Simply speaking: you are correct in all points.

A bit more complicated: The equation for DoF has four variables (subject distance u, focal length f, circle of confusion c, and aperture setting N). Only c is format dependent, and all but f are linear (see attached equation).

The picture comparison illustrates the increase in DoF with smaller formats.
Picture 1 shows that a 24mm lens has the same viewing angle with a small format as an 85mm lens with a larger format. Picture 2 shows that, if enlarged to the same image size, the smaller format has a DoF advantage at the same aperture setting, due to the shorter focal length.

Hope this helps.