Yes, power output can be regulated in nuclear energy. It is, however, not economical to do so most off the time. Building a nuclear reactor is a massive capital investment, so any time you’re not running at 100% you are increasing your payback time, which leads to more expensive electricity.

That technology is nowhere near mature enough to provide a solution to the mentioned problems in the next decade or two.

The EU supreme court or the EU data protection agency, roughly.