Is it possible / legal to gain a copyright / patent over a somebody's else idea, changing / improving / altering something significantly?
This site, and my expertise, is confined to patents so I will not address copyrights.
Patents cover inventions - the structure of an apparatus, the steps to take to achieve a result, the configuration and operation of a system are examples. The patent law's broad list what can count as an invention ends with the words "or any new and useful improvement thereof". Yes, improvements to an existing idea are patentable if they are truly new and not obvious. Most patented inventions can be seen as improvements to something that went before.
OK, you get a patent on your improved dishwasher. That does not mean you can make and sell that improved dishwasher. The older, existing, dishwasher you are improving might have patents on aspects that the improved one you make and sell also has within it. If I have a patent on the power lawnmower and you add a seat, it is new and patentable by you, but it is still a power lawnmower. Neither you or I can make the improved lawnmower without a deal with the other.
Also - there are myths floating around about needing to make "a 15% change" to something existing in order to be able to get a patent on your new version. Of course there is no precise ruler to measure a percentage of change to an invention and the patent system does not think in that way. Sometimes a small change makes a big, and surprising, difference.