Although the Boomerang may be used in order to cut down the ropes to Puppet Ganon's tail, the common speedrun strategy (especially since the Boomerang is usually skipped) is to snipe the blue ball on his tail. In order to perform these snipes successfully, you should time your arrows based on the tail's movement. Shooting arrows when his tail starts to slow down can also be a successful strategy.
In this Phase you just hit his Orb intentionally. However, there is a strat you can do, which allows you to hit the Orb twice within one "phase". As soon as he falls down, you want to stand on the left side of his Orb and shoot it. After that you hold down-left on your controller and aim for the Orb while it is spinning. It can be tricky at first but you will get it down pretty quick.
There are three possible strategies for this phase. The first is to attempt to snipe his tail, which can be achieved consistently with good timing and practice. The other strategy is to run into Puppet Ganon's head with a bomb, causing him to stop, making it easy to hit his tail with a Light Arrow. Since Puppet Ganon's movement is random, it is sometimes difficult to hit Puppet Ganon with a bomb if he does a bad movement pattern. However, the first Hit is really consistent, because the way he moves at the beginning is always 1 of 3 patterns. 2 of these patterns can do a quick snipe in the exact same way. After this first hit, you want to make him stop to perform a mid-air snipe which can be set-up by standing straight behind Puppet Ganon.
A commonly discussed trick in The Wind Waker is the Puppet Ganon skip, a very hard and luckbased trick where you zombie hover up until you are above the entering to the final fight and rely on a bomb drop to kill a keese and spawn a heart to survive. Sounds difficult? That is because this trick is difficult.
To skip Puppet Ganon you first need to zombie hover up until you are above the entering of the final fight with Ganondorf himself. To succeed your hover you are required to press the button with a speed of 13 inputs per second or more to even be able of reaching the top. With less speed than that, you will just find yourself getting knocked back by the keese that spawns in the chamber. Even if you would be able to actually get into the loading zone to the final fight, it would be no use since you would die after the cut scene and respawn back down in the tower.
After you have reached the top the next step is to drop a bomb during your hover. You can not drop bombs in a zombie hover by pressing B or throwing it, and the only way of releasing the bomb during the hover is to take out your deku leaf. This will replace the bomb Link is holding and you can continue hovering right after (since it is not possible to leaf with zero hearts). The bombs themselves has to kill one of the keese attacking you, which will spawn a heart. Even if this trick, up to this point, is a lot about skill level and skills only, the placement of the hearts are completely random and you will have to actually touch one to survive the hover.
When the hover is done you can just drop down to the floor beneath you and roll into the loading zone.
The Tingle Tuner (an in game item) can not be used in rooms without a map and therefore not in boss rooms. This is the exact same reason why Helmaroc King can not be skipped in the same way.