Q:

Are focus and resolving power optically equivalent?

Limited "resolving power" or "sharpness" is a problem one can see with many lower-quality lenses, especially with higher-resolution camera bodies.
Playing with my cheap 500mm mirror lens, I often get images like the following crop in which I slightly miss focus on my subject, but even the objects right at the focal plane are "blurrier" than what my camera's sensor can pick up.

Optically: for a particular point in an image, is "low resolving power" the same as "blurred focus?"
E.g., do both produce a gaussian distribution of an imaged point?  (The difference being only that "focus" varies the blur depending on the distance to each point in scene, whereas "resolving power" blurs uniformly across the scene?)  Or is there something quantitatively different about the blur that one gets from focus, as compared to the sharpness one loses with inferior optics?

A:

No lens (even a theoretically perfect copy of the intended design) with real thickness focuses all the light that enters it at the same distance.¹ So if some of the light reflected from a specific distance is in focus, some of the light reflected from the same distance will be focused slightly further or slightly closer to the lens.
Although the primary and secondary mirror surfaces of a mirror lens has no real thickness because the reflective substance is applied to the front of the mirror, the vast majority of mirror lenses also have refractive elements in the optical path. There is also the issue that just as with refractive lenses, the mirrors in such lenses are not manufactured to the perfect theoretical shape of their design. The higher end mirror lenses once offered by the like of Nikon and Zeiss came much closer than the current low end fare does. Rather than having parabolic mirrors that are more difficult and expensive to make well, most mirror lenses have spherical mirrors that are combined with an aspherical front corrector plate that also serves to keep (most) dust out of the lens and to support the secondary mirror in the middle of the front of the lens. Many catadioptric telescope designs use a similar optical formula. 
Well corrected lenses are able to focus most of the light reflected from a specific subject at a specific distance over a smaller variation in distances than less well corrected lenses. What we call the 'point of focus' is when we have the lens positioned so that light from a single point source is projected onto the imaging plane in as small a blur circle as is possible.
This blur circle is often called the circle of  confusion. If a CoC is small enough, it looks like a sharp point to our eyes. As the CoC grows larger, eventually it becomes large enough that our eyes can tell that it is not a single point. This is why things that may look sharp on a 4x6 print of an image may look very sharp, but the same things in a 16x24 print of the same image may look blurry when viewed from the same distance. The second print is enlarged 4X the size of the first, so the size of the blur circle, as projected on the sensor or film, must be 1/4 as large to tolerate the 4X greater enlargement. 
In the modern digital era, pixel peeping has forced the conventional ways of calculating acceptable CoC into obsolescence. When you look at a 24MP image at 100% magnification on a 24" HD monitor, you're looking at a piece of a roughly 60x40 inch enlargement! That's a far cry from the standardized 8X10 inch print viewed from a distance of 10-12 inches upon which most CoC calculations are based.
As the diameter of lenses increases, the difference in focus distance between light rays from a single point source of light striking the center of the lens and light rays from the same point source striking the edge of the lens are focused at increasingly larger differences in depth. The difference between light waves at different wavelengths also increases. Even cheap mirror lenses tend to do very well at controlling chromatic aberration because the refractive elements in the optical path are of relatively low refractive power. But they tend to struggle with focusing all of the light from a single point source striking different parts of the lens to the same depth behind the lens. The size of the smallest blur circle a lens can produce from a point source of light determines, to a large degree, the lens' sharpness.

Optically: for a particular point in an image, is "low resolving power" the same as "blurred focus?"

The effect is similar for scenes of fairly uniform brightness containing no specular highlights but the nature of the blur circle from each point source of light will often be different for the two cases.
If the cause of the blur is missed focus with a well corrected lens, the blur circle will spread the light striking the edges of the lens in much the same way as it spreads the light striking the center of the lens. The blur will be fairly uniform in brightness and color from the center to the edge of the blur, with the center being brighter and the edges being dimmer.
If the cause of the blur is a "low resolving power" lens, as the lens' focus is moved slightly some of the light striking one part of the lens can actually be brought more into focus as some of the light from the same point source striking other parts of the lens is blurred even larger. This results in blur that is not as uniform. The edges may be brighter and more distinct than the center, or the blur may be shaped in non-circular ways, such as what we call coma. With mirror lenses this is exacerbated by the blockage of the very center of the lens that does not allow any of the most collimated light from point sources at less than infinity distance from entering the lens at all. This makes the blur of out of focus objects look like donuts with a hole in the center.
¹ Perhaps a theoretically perfect single wavelength laser beam shot perfectly through the center of a converging lens' optical axis might, but then how would one tell if such a single beam of light were out of focus? In either case it will only strike a single pixel well.

