In my humble opinion only the first one of your statements is true. Indeed it is true also for a DX camera. It is well known that larger pixels catch the light better, seem to have some advantage when it comes to noise and sharpness seems to improve. This is what I know, I could be wrong but the D2H I owned told me a lot about the number of pixels and sharpness. It only had 4 megapixels but I dare to say that the enlargements from that camera were of superb quality.
I do not know if you will find this information relevant but a DX camera using a DX lens uses the whole sensor as a full frame camera does. I have always advised those "updating" to a FX body to use only FX lenses. Makes no sense to buy a full frame to use crop lenses.
By the way, I use a D610 and a D7000. I would be a fool if I admit that I see a difference in the quality of the images between both cameras. Both do the job when I do mine.
Wide angles used to be an issue with the crop sensor till Nikon introduced the 12-24 f4 AF-S and from there on wide angle photography with a crop sensor is no longer an issue. When it comes to teles we all know the crop sensor does better.

That is like saying the Earth doesn't have an AU factor. It does — ONE. (one Astronomical Unit = 149597870700 meters, roughly the average distance from Earth to Sun)

Using FF as a reference point of ONE is no longer as meaningful as it was a decade ago.

It was initially ONLY a crop factor because early dSLRs were just 35mm SLRs, highly modified. Kodak and others stuck a small sensor in the middle of the film plane of a Nikon or Canon, drew a rectangle on the focusing screen, and called it a digital camera. There was a "crop factor" only because the projected image cone of the lens was cropped. Photographers needed some way to equate the change in magnification from a lens on full frame to a lens on a smaller sensor, so they could choose the right lens for a scene.

A more appropriate term would have been magnification factor. Modern APS-C and Micro 4/3 cameras, used with native lenses, DO NOT CROP the projected image cone, because the lenses are designed for the format. But put a full frame lens on an APS-C body, and yes, you can say it has a crop factor… and a magnification factor.

The distinction is not just a matter of semantics, but a matter of the physics behind lens design. All other things being equal, a lens specifically designed for a given format can be optimized for that format. That APS-C and DX NATIVE lenses are often built to lower standards than their full frame equivalents is just the hubris-blind marketing of the major dSLR makers.