I want to split the paths at each intersection so that I can delete the useless segments and merge the remaining ones. How can I do it?
I would abandon the method you're trying to employ, because the gears won't fit if you try to do it that way.
The tooth size of gears must be the same so that the gears will fit together. It's the number of teeth that changes for smaller gears.
The best way to do this in Inkscape is to use Extensions > Render > Gear > Gear.
Make sure you keep the Circular pitch (tooth size) the same each time you render a gear. To get smaller gears you need to reduce the number of teeth. The beauty of the Gear extension is that it does all the maths for you!
By using Path > Cut Path you can split with the topmost path one underlying path when both are selected before applying. The path to be splitted can be a combination of multiple paths (Path > Combine). The cutter path vanishes, so duplicate it, if you need it.
Consider to fill with the paint bucket tool what you want to keep and delete the strokes after that. Filling creates new shapes that can have strokes, too, if you want to fill the gaps.
Not asked, but still: Gears in mechanical machines have about 200 years had complex forms which are mathematically designed for continuous transfer of torque and keeping tear and wear in minimum (=no peaks in surface pressure during the revolution). Circle sectors are not up to the task. They are in practice impossible even to make fit. If you want your own teeth patterns, consider to use path function named "Pattern Along Path". Careful calculations are still needed to make all fit.