Paying cumulative upkeep is always optional. If it’s not paid, the permanent with cumulative upkeep is sacrificed. Partial payments of the total cumulative upkeep cost can’t be made. For example, if a permanent with “cumulative upkeep {1}” has three age counters on it when its cumulative upkeep ability triggers, it gets another age counter and then its controller chooses to either pay {4} or sacrifice the permanent.

This ability has no duration. An affected creature will retain the triggered ability until the game ends, it leaves the battlefield, or some other effect causes it to lose its abilities. It doesn’t matter if Musician is still on the battlefield or not.

Although a creature can gain the triggered ability only once, it can be targeted by multiple Musicians to increase its number of music counters.

If a music counter is put on a creature by some other means (Fate Transfer, for example), but that creature has never been affected by Musician’s ability, that creature won’t have the triggered ability and its controller won’t have to pay mana. The music counters will stay on that creature but won’t do anything.