No, they do not. The lens creates an image circle, regarless of what sensor (or if there is a sensor) behind it. That image circle includes the field of view as well as the depth of field.
A FF sensor 'crops' the image circle to a FF sized rectangle. The APSc sensor 'crops' it to an APSc sized rectangle (taking only the middle section of the full image circle of a FF lens). So you're getting a smaller FOV, as if the lens were longer in focal length. The image scale difference, or 'reach' as people like to call it, comes from the higher pixel density that is typical with APSc sensors vs FF sensors. So the Duck is the same physical size in the image circle whether the lens is on FF of APSc. The space around the duck is less on APSC, since it is not capturing as much of the image around the duck (Crop factor). The number of pixels per duck is higher on APSc (typically), meaning the resulting image of the duck when viewed at 300ppi (for example) is larger. This is 'reach' advantage of APSc.
Depth of field in the two images is the same. Where it would change is if you tried to match the FF field of view on an APSc sensor by going to a shorter focal length lens. For example, a 50mm FF image has about the same FOV as a 35mm APSc image. If both are taken at F2.8, the FF will have shallower depth of field, since the thickness of a focal plane at 50mm f/2.8 is thinner than a 35mm f/2.8 lens. If you are using the same lens at the same aperture, the DOF in the resulting image will be the same on both sensors (but the APSc sensor image will show a smaller FOV, as described above).
So on an R7, the effective FOV of the 100-500 is 160-800 (crop factor). The image scale difference ('reach') will be about 50% greater (more pixels per unit area) than an R5. And the DOF will not be affected.
Hopefully that helps.
Brian