Noun + participle as adjective phrase

First, hobo-slept doesn’t work because (in most dialects) you can’t say “I slept the bed”. But I think there are a couple of different things at work in your other examples. In the first set, the noun part of the compound modifier is pretty much required in order to make sense of the sentence:

This is a built home. (Built by whom?)

Our newspaper is run. (Run by whom?)

The beaches were littered with soaked birds. (Soaked with what? Water?)

Whereas in your other examples, the subject can be assumed:

This is a used litterbox. (Used by a cat.)

This is the most wanted toy. (Wanted by anyone.)

But for those cases where there is a common assumed subject, you can change it explicitly:

This isn’t just a used litterbox—it’s a human-used one!

As far as I can think of, that’s only done for effect. It’s no wonder, then, that child-wanted sounds a bit odd when you aren’t deliberately contrasting it with, for instance, another kind of person doing the wanting. I would say the general rule is this: if the subject part is not required, then it can only be included for emphasis.

A couple of differences occur to me:

• in e.g. student-run, expert-administered etc, there's a notion that the agents act together as a group; this isn't the case in !child-wanted toy, !parent-bequeathed house etc;
• possibly related, another difference is that in the cases where the compound is readily used, it seems to be the case that the recipient isn't one of the arguments of the verb.

Or maybe put more simply, the construction works better when there's a notion of "altruism".

However, these are really just the differences that occur to me off the top of my head-- I'm also sure it's more complicated than this.