Technically speaking, "tidy up" would be a little more accurate for that scenario, but I would say that it's more common in some regions to say "clean up" instead. In daily usage, clean up doesn't strictly mean removing dust or dirt. In Canada, I never hear people say "clear up", so I think that's also a regional difference.

The word you want here is 'clear up', rather than 'clean up'. 'Clean' suggests the removal of dirt. If you are just tidying things away, you are 'clearing' them up.

I'm not exactly sure about this, but I think that tidy implies being neat. For example a kid can arrange the toys in a line on the middle of the room, now it is neat/tidy, but not clean. Whereas to clean is to completely remove the toys from the floor and up to the shelves or the storage areas. I think "clear up" works fine here too. Bare in mind some words are not perfectly sensible in their current usage, cause it was originated for different uses than we might be using it for now for example "Punk", google it :P