Dorfs clean on constructions outside if the options are set up to do so, but not dirt. They clean on dirt indoors.
To recall a cavern cleaner you'd probably have to activate a civilian alert (a normal burrow will probably either freeze the bugger in place unless a job can be found within it, or have him walk all the way to the location and then give up).
Edit:
Finally reached the stage where I'd built a roof over the whole area, and tried a number of things (save scumming):
- Magma: Doesn't work.The vegetation burned, but the muck remained (and the fire spread outside of the containment).
- Dirt roads: Might have worked to some extent. Almost all of them got suspended because the huge piles of stuff on them, however (100? dead visitors cluttering the area with their stuff, plus a number of animals). However, it looked like the goo was cleared away underneath many of them.
- Cleaning: To my surprise they do clean dirt outside, and they did so fairly vigorously, including under dirt roads under construction (which is why I don't know if construction itself has any effect). I ended up trying dirt roads, cancelling most of them, and adding a few dirt roads targeting tiles that hadn't been cleaned.
- Floors: Dirt roads can't be built on rock (including muddy rock, unfortunately, so I don't know of any good way to get rid of vegetation so it can be smoothed away), so I've added a small number of those.
- A dwarf washer at the entrance showed it being contaminated and the contaminants being washed away, so the bugger still drag the muck around.
- I've yet to see if I get any casualties of the operation, as the effects are insidious. Looking at affected visitors you see them covered in muck, with wounds showing nothing, but a general look at them shows blisters everywhere, with destroyed tissue. What gets living creatures seems to be suffocation due to blistered lungs. The two undead villains teleported into the area eventually died, but it took a long time, and I don't know the cause, while the experiment one apparently was killed by a were.
As an aside, getting a dwarf washer to work with a slow aquifer is a pain...