Once Acrobatic Cheerleader's ability triggers, it won't trigger again as long as it remains on the battlefield, even if the triggered ability is countered or the flying counter is somehow removed. If an effect would cause Acrobatic Cheerleader's ability to trigger again, that part of the effect won't do anything.

Survival abilities (and other abilities that trigger at the beginning of your second main phase) will trigger at the beginning of the second main phase you take in a turn. They won't trigger during your third, fourth, or other additional main phases in a single turn, if effects somehow cause you to have more than two main phases.

If a creature's survival ability triggers but the creature leaves the battlefield before the ability resolves, use its tapped or untapped status as it last existed on the battlefield to determine whether or not the ability will do anything.

If a creature's survival ability triggers but that creature is untapped when the ability begins to resolve, that ability won't do anything.

If a creature with a survival ability isn't tapped when your second main phase begins, the ability won't trigger at all. You won't be able to tap it during your second main phase in time to have that ability trigger.

If Acrobatic Cheerleader leaves the battlefield and then returns, it's a new object with no memory of its previous existence. Its ability will be able to trigger again.