When the ancient one awakes, what will happen with the another monsters? They stay in the board or just they're banished? And if they stay in the board, the player needs to evade that monster every time, or just one time or never?
The another question is about the dice success, every FAQ and fourum say the number of cumulative success of dies but, what happens when in one whole round is just one success? It accumulates to the next round or it's just lost.
During the final battle, the only thing that matters is the Ancient One and any Investigators who were not lost in time and space. Everything associated with the old board is irrelevant.
Take a look at the description of the turn structure on page 22 of the rules:
Combat with the Ancient One is divided into rounds. During a round, players first have an opportunity to adjust their investigators. Then each player attacks the Ancient One. Finally, the Ancient One attacks the investigators. Once these steps have been resolved, a new round begins.
Where in there would monsters attack? And Investigators may only make combat checks against the Ancient One.
Your second question is answered by another paragraph on page 22:
Instead, keep track of every success an investigator scores against the Ancient One (see "Skill Checks," page 13). These successes are cumulative, and each successive player adds to them with his own attack. When the players accumulate a total number of successes equal to the number of players (including any players that were eliminated from the game), remove one doom token from the Ancient One's doom track and reset their cumulative successes to zero.
And further clarified by the FAQ referenced in this question.
If the investigators get 9 successes in the first round of combat, they would remove 2 doom tokens, and 1 success would carry over to the next round of combat.
