How to prevent the derailleur cable fibres getting all over the places?
You can buy cable caps separately, they're actually meant to be crimped on so you can fix them with a crimper or pliers.

I know Park tools do something which is basically a really sharp wire cutter, should prevent fraying in the first place. This tool is showing on their site now but I have one (bought many years ago) which looks a bit different.

You shouldn't need to resort to duct tape!