What do you do about tire dust? Do you just cover the area of wall behind the trainer?

I just have my trainer on a mat and if it gets bad I just wipe it with a cloth or vacuum it.

I use a trainer tire and never notice any.

That too. Different tires wear down faster than others.

This is just one of many reasons why a trainer tire should be used... A normal road tire is designed to be supple and provide good grip and wear charateristics on a rough surface, where you have minimal amount of heat generation. On a trainer, you have a fairly large contact surface on a smooth surface, which will generate heat (which is the lead cause to tire breakdown on a trainer) and as an added bonus, will provide sufficient grip on the smooth surface resulting in much less noise.

... and a trainer tire is not very expensive...

Trainer tires are a myth. I have one of those yellow conti tires on my computrainer and there's yellow flakes all over.

I've logged a lot of time on a trainer over the years, using the same Conti 1000 tires I use when I ride outside, and I've never noticed tire dust.

granted that my trainer is in my unfinished basement, keep a large portion of cardboard under the whole thing on the concrete floor, it collects plenty of tire dust, and also lube overspray from my chain.