As OP has already established, there's a phrasal verb to water down (literally and figuratively, to dilute). But that's not relevant to the cited usage here, where it simply means to [make] wet. Per my comment, this is clear from the context, which explicitly says that the roads were dampened to reduce the dust.
OP is correct in supposing that down in this instance simply refers to the fact that the water is splashed down[wards] onto the road. But I think it's fair to say the semantic content of the word is negligible here, since the meaning should be just as clear even if it's omitted.
Taking a different verb, it can be seen that [optional] prepositions are often added after common verbs of action where the [minimal] semantic content of the preposition can be largely ignored...
He ate his tea in a hurry, brushed up his jacket, and started off.
The man lifted himself off of the ground, brushed off his jacket and walked away.
He stood before the mirror while his valet brushed out his jacket.
In those examples, native speakers would be unlikely to register any difference in meaning if those prepositions were transposed - or more importantly, if they were omitted altogether.
In my examples, to brush up can sometimes be a "phrasal verb" (to review; refresh one's memory) - but that clearly doesn't fit the context, so it can be discounted.
My advice in such contexts is to consider the possibility of a phrasal verb usage, but if that doesn't quickly suggest a credible interpretation, assume the preposition isn't particularly significant. Often it'll be an almost random choice (but don't add such prepositions yourself unless you're familiar with the usage).
Literally it refers to "checking out of a hotel." It's something you do when you leave the hotel. Once you've checked out, you're gone.
This can be used figuratively in a few ways. The basic metaphor is that once you've checked out, you're gone. It could mean several things:
Death. It's what you do when you leave this mortal life behind. Once you've checked out, you're gone.
Senility. If you're still here physically but mentally you've checked out, then your body is still here but your mind is gone.
In your example, it's like #2 but perhaps without the sense of permanence. Physically speaking, you're sitting in class or you're at a meeting, but mentally speaking you're not there at all. It doesn't appear to suggest actual senility in this case, though—it's just being used to describe a lack of attention. This seems to me like a slightly more extended use of the metaphor.
Best Answer
To walk [someone to a place] means to escort or accompany them to that place; to walk with them.
In this case, if you've just fired a man from his job, it's a good idea to have someone walk with him to his desk while he gathers his belongings, and then walk with him to the exit, to make sure that he doesn't steal or damage anything on his way out.