<A HREF="http://stackoverflow.com/questions/111307/whats-p-np-and-why-is-it-such-a-famous-question">$P=NP$</A>

let me tick the list:

1) most likely false, because, as Scott Aaronson said *"If P = NP, then the world would be a profoundly different place than we usually assume it to be."*

2) yes, it's The Open Problem in computational complexity theory

3) ...

4) it's important, again quoting Scott because if it would be true, *"there would be no special value in "creative leaps," no fundamental gap between solving a problem and recognizing the solution once it's found. Everyone who could appreciate a symphony would be Mozart; everyone who could follow a step-by-step argument would be Gauss..."*