1) (a) implies that there is an established best that this person is even better than, (b) is a claim that the person is better than all others
2) In many cases, they can be, but 'better than the best' is poetic whereas 'he is the best' is straightforward
3) I would say yes, (a) always implies (b), at least speaking logically. One might say something like 'he's not the best... he's better than the best' to be poetic though.
4) I don't think 'better than best' is a common phrase without having 'the' in there, but 'better than the best' is a relatively common expression. The reason is that here 'the best' refers to an understood best or a presumed best.
"Who's the new guy?"
"That's the man who's going to win the tournament."
"This tournament has the best players in the world in it!"
"This guy is even better than the best."
If you were spelling it all out literally, what's actually being said here is that there is a new best.
Sometimes it's also used to indicate that there is something qualitatively different that is superior.
"These cigarettes are the best."
"You should try not smoking, it's better than the best cigarette."