Do constant-aperture lenses make a practical difference for low-light shooting, in aperture or sheet speed priority mode? If the aperture changes as you zoom in with a variable-aperture lens, the camera can always compensate by varying the shutter speed and / or ISO, so does it make a noticeable difference in practice, for low-light shooting?
Many times when shooting action in limited light you are using manual exposure mode with the widest aperture and the slowest shutter speed you can without getting camera blur or blur from the motion of your subject(s). By using a constant aperture zoom lens you don't need to change the ISO to compensate for the change in aperture as you zoom. Many cameras allow the use of Auto ISO in manual shooting mode but some don't. Even with Auto ISO, the way some cameras handle "partial stop" ISO settings (i.e. ISO 125, 160, 250, 320, etc.) make them less than ideal compared to "full stop" ISO settings (i.e. 100, 200, 400, etc.). In such cases the preferred way of using Auto ISO is with "full stop only" ISO values enabled, which means as a variable aperture lens moves through maximum apertures in 1/3 stop increments the exposure "bounces" up and down. For what I shoot on a regular basis there is a world of difference between a constant aperture zoom and a variable aperture one.
Would I get a lens of better quality for the same price if I don't insist on a constant aperture?
In general constant aperture zooms tend to have better optical quality than variable aperture zooms. It is not so much an inherent quality of a constant aperture design as it is an indicator of what the market demands out of higher priced lenses. Usually, but certainly not always, the tradeoff isn't better optical quality in exchange for a variable aperture design. Rather it is variable aperture in exchange for a cheaper price.
And, taking a step up from F4 constant-aperture zooms, how do F2.8 constant-aperture zoom lenses compare? Are they generally as sharp as prime lenses? In other words, if I don't need a wider aperture than F2.8, would an F2.8 constant-aperture zoom lens substitute for multiple prime lenses in its focal length?
That all depends on the particular lens. The Canon EF 70-200mm f/2.8 L IS II is one such lens that can answer (a qualified¹) yes to your last question. A few other zooms are very close or equal to their prime counterparts at the same aperture settings. But there are other zooms whose image quality falls far short of their prime or narrower aperture counterparts. The Canon EF 16-35mm f/2.8 L II is one such lens in this category. The EF 16-35mm f/4 L is better at pretty much every common combination of aperture and focal length than the first two versions of the 16-35 f/2.8 and is also quite a bit more affordable. The newly released EF 16-35mm f/2.8 L III, however, is significantly better than all of Canon's previous 16-35mm zoom lenses. It's also quite a bit more expensive than the others.
¹ As Roger Cicala has pointed out more than once in his blog series at lensrentals.com, the lens-to-lens variation between all zoom lenses, even the most expensive ones, is much higher than between even mid-grade prime lenses. So a "good" copy of a very good zoom such as the EF 70-200mm f/2.8 L IS II or EF 24-70mm f/2.8 L can be pretty close with regard to measurable I.Q. to prime lenses in its focal length range. But the less measurable characteristics of any two lens designs will vary from one to the next. How the out of focus areas are rendered, for example. Even two very different prime lenses, such as the EF 50mm f/1.2 L and the EF 50mm f/1.4 differ more on things that don't show up on a flat test chart than they do with regard to absolute resolution, geometric distortion, astigmatism, coma, etc.