Wider angle lenses tend to demonstrate more of all of the optical aberrations than narrower angle lenses do due to the need to have higher refractive power to provide a wider angle of view. All of the classic aberrations are caused by refraction, so it only makes sense that the more a lens must refract the light that strikes it, the more severe each of the aberrations will be and the more correction will be needed to control the higher amounts of those aberrations caused by that higher refractive power.
The causes of vignetting with wide angle lenses are at least two-fold and sometimes more:
- Light striking the front of the lens from the edges of the FoV are at wider angles than light from the center of the FoV. If the light from the edges of the scene is the same intensity as the light from the center of the scene, the angles with which the light from the edges of the FoV strike the front surface of the lens will spread that same intensity of light over a larger area of the front of the lens. It's the same principle that contributes to winter when the Sun is lower in the sky over the course of shorter days and the angles of the Sun's rays as they strike the Earth mean the same amounts of energy are absorbed over larger areas of the Earth's surface in the hemisphere where it is winter than in the hemisphere where it is summer and the Sun is higher overhead.
- More light from the edges of the FoV of wider angle lenses is lost than from narrower angle of view (longer focal length) lenses due to the higher refractive power such lens need to bend light for wider angles of view. Light from the edges of the FoV that passes through the lens' glass elements at a steeper angle must pass through more glass than light that strikes the front of the lens head on. The more glass the light striking the front of the lens passes through, the more of its energy is absorbed by the lens.
- In the case of unevenly shaped apertures, there is also either mechanical vignetting caused by something in front of the lens' front element or vignetting caused by the lens elements themselves not allowing the full entrance pupil to be seen from all parts of the lens' FoV. In the former case that something is usually a part of the lens' barrel. In the latter case it's usually a consequence of designing the front surface of the first element of the lens with a relatively flat surface instead of a highly bulbous one.
Does this lead to any practical gain in depth of field in corner areas?
Not really. The reason smaller apertures increase depth of field (DoF) is because they prevent less collimated edge rays striking all around the outside of the front element of the lens from passing through while allowing more collimated center rays to pass. In the case of vignetting, though, the light being reduced is not all edge rays - along with edge rays from the other side of the lens' front element, some of the light lost is closer to the center. The light allowed through the lens isn't all more collimated center rays, either. Much of the light allowed to pass through is light that strikes the nearer edges of the lens' front element, too.
Does it add to field curvature in lens designs that already have focus shift from changing aperture?
Quite the contrary. Focus shift is partially a consequence of uncorrected field curvature, not the other way around. Vignetting does not cause field curvature, either. Vignetting is a consequence of the same thing (a wider angle of view requires more refraction than a narrower angle of view) that also makes correcting field curvature, which is already there, harder to do in wider angle lenses.
Is the unevenly shaped aperture (longer than it is wide) in the corners expected to cause any astigmatism-like effect (or actually explaining astigmatism)?
Again, correlation does not mean causation. Astigmatism at the edges of wide angle lenses is a result of the first two things listed above that cause vignetting: higher refractive power and light from the edges of the FoV striking the front of the lens at wider angles than is the case with narrower angle lenses, as well as the different angles light from a singular point near the edge of the lens' FoV will strike different parts of the lens' front element.