Is there a version of Illustrator's "Divide Objects Below" for Inkscape?

If I understand what you are asking, this is what you want to do.
you have a bunch of paths creating an image and you need to slice the image with a line.

so here is an image and we want to cut it on that black line.

And we want this (two movable parts).

To do this:

1. You should group all your paths.
2. Duplicate and move it (you will need to move it back so take note of where it was).
3. Draw your line.
4. Select your line and use the pen tool to extend the line into a rectangle or if you need a straight line then just use the rectangle tool over the part that you want to keep.
5. Duplicate the rectangle and move the points to the other side that you want to keep.
6. Now select your image and then your path, go to Object > Clip > set
7. Move the other image back and do the same.

Now if you just want to divide a path from another path.

Here are two path objects.

And an example, I made the top path transparent so you can see under.

you just go to Path > Division.

Or if you need to cut the difference of the top path and the path under.

Path > difference

Or, except where the top path and bottom path intersect.

Path > Exclusion

If you need something else just replay!