I'm considering buying an old manual focus prime lens for my DSLR.

Be aware that you ( at best ) will lose not only autofocus but also auto metering ( you'll have to set exposure manually ).

Be careful as well as not all DSLRs can mount manual lenses properly.

I've noticed that they're often listed very cheap if they're described as "slightly clouded" or "hazy". I would like to understand whether this is a problem I can work with, or if I should steer clear.

There is a problem with this.

It's sign that mold ( fungus ) has started to develop on the interior of the lens. I would never buy such a lens. It's (in general) not fixable and it just gets worse. Mold can ( will ) eat at optical coatings and even such attempts at cleaning as I've seen over the years will not undo the damage.

In my experience patience will let you find good lenses without mold for similar prices ( maybe a tiny bit more ), but any lens with even the suspicion of mold is not worth having, IMO. From time to time you may find a lens you already own develops mold, and there are techniques to try and keep this manageable, but why buy into this problem ?

Secifically, my questions are:

What are the effects of clouding or haze on the resulting images ?

Varies depending on the extent of the mold. The problem is that once you have mold it's going to get worse.

What causes clouding or haze in a lens?

Mold ( fungus ). And it's very persistent. While I think of it, try and keep your own lenses stored with desiccant packs or similar to keep humidity down to avoid this issue.

Does it affect all images, or does it only manifest under certain conditions? (I.e. can you mitigate the effects with shooting technique?)

Varies depending on how bad it is and exactly where in the lens it is.

Can the effects be fixed in post-processing ?

Not really.

Is the effect of a hazy lens always bad ?

Yes.

If not, are there examples where it has been used creatively to good effect ?

I've seen people use smashed up lenses for "creative" effects. This is quite different from using something normally ( and something you paid for ) and expecting only to use it for "creative" effects.

If it makes a difference, I use an APS-C DSLR and I would be buying a ~50mm manual focus fast prime, for street photography and as a general walk-around lens. I'm a hobbyist and mostly want it as a first step into trying manual focus

but I do want to produce high quality images.

Well avoid cloudy lenses for a start !

But also note that a manual fast prime will not automatically get you high quality images ( no lens will ). All a fast prime gets you is a wide aperture. All manual focus gets you is a requirements to learn to focus accurately yourself. Note that it was common for film cameras to use a type of viewfinder called a split focus screen which helped get accurate manual focus. DSLRs do not normally have these and manual focus can be a pain without one. These days I manually focus using a mirrorless camera that has focus peaking and an EVF, which is better.

So ask yourself exactly what you expect an old manual prime to do for you and is that realistic.