UPDATE: Since I posted this answer, some extension scripts for multiple path operations have been posted on the inkscape forums: http://www.inkscapeforum.com/viewtopic.php?f=5&t=20762
Use these at your own risk; they are provided as an unofficial temporary solution until Inkscape devs work on this feature!
This is not a perfect solution, and it does not suit all situations. However it’s the nearest thing to Illustrator’s “divide all objects below” operation that I could find. If anyone knows a better way please share.
A group cannot be divided, as you know. However multiple paths can be divided if combined, along with "exclusion" (at the expense of losing individual styles/colour).
- Select all the paths you wish to divide and go to Path>Combine
- Select your combined paths and Edit>Duplicate
- With this duplicate, go to Path>Break apart
- Then go to Path>Exclusion to create a path that will divide paths below
(I've changed this Exclusion layer to a darker blue to make this clearer to follow)
- Select all paths and this exclusion path then Path>Divide
This should divide all paths from your original shapes.
Note: Sometimes using exclusion with overlapping shapes can create gaps, and make it unsuitable as a "divide all below method".