Yes, it's one of the major issues with Youtube right now, even with then putting funds into escrow. It's also an opportunity to have fun, such as people uploading videos to deliberately trigger DMCA claims from multiple parties so that they can fight over it.
Not if there were simple measures put in place to stop new or otherwise unused accounts from making copyright complaints. Just to be thorough there could be an outlet for unregistered users that had a higher threshold of evidence required before taking the alleged infringing video down.
Kind of funny how it takes more effort to earn the privilege of downvoting on hn than it takes to make a legal statement (perjury yada yada) but I imagine adding these constraints will lead to a lawsuit before you can spell class action. Even if the lawsuit ultimately fails, it costs a bunch to defend it. In the face of effective zero competition, how can they justify the expense/risk to their corporate owners?