An interesting bunch of examples, and correctly grouped.
However, the three groups are not monophyletic. Briefly,
Group A is an example of what linguists call a "rule conspiracy", where a number of independently motivated processes "conspire" to produce a similar surface structure. Georgia Green discussed them in her paper [Green, Georgia M. (1970) 'How Abstract is Surface Structure?' CLS 6, 270-281].
What's come to be called the 'Green Conspiracy' includes such structures as
- I shot him dead.
- I buried him alive.
- I found him alive.
- I need him dead.
et cetera, with very different meanings.
The point, if any, is that there is a limited number of surface structures that English prefers, and there are many more different ways to get from meaning to one of them. I.e, these structures do not represent a single kind of meaning, but rather several. They are all, of course, regular (in much the ways suggested by the OP), but which rule gets used is arbitrary and idiomatic.
EDIT: a little more about Green's paper, which seems to be difficult to find.
This is from a paper by Goldsmith and Huck commenting on the theories involved.
Green (1970), noting that a variety of different semantic structures could be associated with the same surface syntactic construction, argued that there must be a limited set of syntactic “target structures” into which the transformational rules map their
representations. The sentences She shot him dead and They buried him alive, she argued, both share the same superficial syntactic structure, but crucially differ semantically as to whether the adjective indicates a pre-existing state or a result. As she pointed out, “natural language syntax is free to utilize mechanisms by which a large and diverse set of logical and semantic relations are somehow squeezed into a small number of surface structures” (Green 1970:277). In that paper, she referred to such mechanisms as “conspiracies.”
Group B is a conflation of several varieties of Raising and Equi,
with different kinds of tensed and untensed complement clauses.
Group C consists of several examples of the rule of to be-Deletion
(p.9 in the Transformation List).
Best Answer
When the sun goes down is the common use and form for a setting sun (as oppose to the sun comes up for a rising one).
The image in my head is a person standing on the beach in the evening waiting to take a photograph of the setting sun.
Let's wait for the sun to come down seems to imply that you're waiting for the sun to catch up to you for some reason (as if you're ahead of the sun).
The image in my head is a person standing on the beach in the late afternoon and telling his/her friend(s) that they should stay until the sun reaches them and sets in the distance.
There seems to be no formal difference, just a directional thing. I guess if your back is to the horizon and you're not watching the sun there, it's coming to you. And if you're facing the horizon and the sun is there, then the sun is going from you.
On a side note, there are lyrics of songs with both versions.