Q:

Do photographers avoid using zoom?

Because most lenses do no have a fixed aperture, do most photographers avoid using zoom as to avoid messing up their settings and their established exposure? 

A:

do most photographers avoid using zoom

No.
If "most photographers" avoided zoom lenses with variable aperture, there'd be fewer zoom lenses with variable aperture on the market. Furthermore, there are plenty of fixed aperture zoom lenses available at a range of focal lengths, so it's safe to say that photographers don't have to avoid zooms just to have fixed aperture in most zoom ranges.
I suspect that by photographers you might mean professional photographers, but even so, different photographers have different priorities. Are you talking about sports photography? Landscape? Portrait? Fashion? Photojournalism?

avoid messing up their settings and their established exposure?

I don't think this is a huge priority for "most photographers." When you zoom in or out, you're changing the shot significantly anyway; considering exposure again should be part of that process. 

A:

Most photographers will weight various factors to get a particular shot. If you can't get the framing you want while zoomed out, you'll zoom in. But if that changes the depth of field (because the aperture is now f/4.5 instead of f/2.8, or whatever), you need to weigh how important depth of field is to the shot. (Or don't weigh it, take the shot anyway, and see how it turned out.)
It's like any other parameter on a camera. You don't give up on a fast shutter speed just because there's less light. You compensate with a wider aperture or a higher ISO, or by adding your own light if the shutter speed is really important. Or, you try at a slower shutter speed and see if you can reasonably get the shot, or perhaps get a different shot that looks better at a lower shutter speed.
If you want to keep your aperture wide open and your lens doesn't allow it at closer zoom levels, you either move closer (zoom with your feet), zoom in anyway and deal with the larger depth of field, or compose the shot differently. It's all about choices. But I don't think most photographers are avoiding zooming very often.

A:

I think zooms are even more popular than ever with photographers.  The optical quality of even a modern kit zoom is so good that it's often hard to justify using a prime, unless you specifically require a wider aperture.
The "fear of zooms" comes from the period long ago when most cheap zooms were considerable optical compromises for a serious photographer and when even the cheapest film SLR often had a 50mm f2 attached by default.
Those days are gone.
With the combination of really excellent high ISO performance on even the cheapest DSLR and good optics of a modern kit lens, it's often hard to justify using anything else.

