Bonus question: How many names would one need to right down?
Real bad luck some chud hog like Light Yagami found the Death Note before anyone else did. By killing only criminals reported on in the news, he effectively made himself into a weapon of execution that could be freely wielded by the cops at almost entirely their own discretion. I wish the anime would have explored that idea a bit more, but I'll settle for the very pathetic way Light Yagami got caught by a 12 year old, threw a tantrum, and then died. That was unironically a great ending.
So say a communist found it instead. How much damage could this baby do to the global capitalist nightmare we live under?
as soon as a new President is elected, write the name down. how long can they keep it going.
If you don't change things fast enough, one problem you could run into is politicians assuming pseudonyms for public life since the Death Note requires you to know both a person's face and real name. That puts the user under a bit of time pressure, but unless you're really obvious you could probably go quite a ways before they find out about this requirement.
What counts as their name? Their birth name? Their legal name? The name people call them buy? Basically; What social sciences department did the gods of death get their masters degree in?
I think it's their birth name because it's clear neither Mello nor L ever actually went by the birth names used to kill them. The Death Note rules are full of arbitrary things like this, most notoriously with the seemingly random time limits on different processes. I guess the king of death just had a very practical mindset in coming up with the death note rules.