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?
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.