If I want to shear an object in half, say, a CTRL+G composite object, I can use the Venn Diagram
tools with an object behind the desired object.

I need to go SHIFT+CTRL+C on the aggregate object, to turn it to a path outline,

but I get the status bar message:

"No object to convert to path in the selection"

What am I doing wrong?

Do I just need to be using the "Edit paths by Nodes" tool?

what if instead of ctl-c on the aggregate, you do ctl-c on each of the objects, (you need to ungroup first)
and then select them all and union them together? ctl-+ ?
then you can cut that in half, no?

Is it possible to to convert an object group to one Path,
and another object group to another path,
and thus aggregate groups?

<<
to convert an object group to one Path
>>

i think the answer is NO.. . . . you can convert an OBJECT TO PATH, but not a group of objects.

why is that? well, if you have a red ofill bject and a blue fill object, and you group them. . . . so far so good.
if you then try to convert that to one path. . . . what color fill would the ONE resulting path be?

the boolean operations only work on paths although some will work on more than two paths. if you union them first then you lose all the distinct fill/strokes. if you want to maintain the colouring (fill/stroke) of the group elements then ungrouping and cutting them one by one is the only way to use a shape to cut it.
the eraser tool can be realy useful sometimes (and quite pigheaded sometimes). deselec everything and then try to draw a line through your group with the eraser set to its 'venn diagram' setting. It might let you do what you want.