The ability Harmonic Sliver grants itself triggers when it enters the battlefield.

The controller of the Sliver that enters the battlefield controls the triggered ability for that Sliver and chooses what to destroy, not necessarily the controller of Harmonic Sliver.

Abilities that Slivers grant, as well as power/toughness boosts, are cumulative. However, for some abilities, like flying, having more than one instance of the ability doesn't provide any additional benefit.

If the creature type of a Sliver changes so it's no longer a Sliver, it will no longer be affected by its own ability. Its ability will continue to affect other Sliver creatures.