It is possible to get a jump off of the pipe connecting to a pillar, this is slightly faster than having to wait for the lava to fall back down. However this is highly inconsistent and isn't recommended to be done in runs.
There are two ways to skip these flame jets. Instead of turning off the first two flame jets, simply walk into them and take damage, then quickly roll through them while temporarily invincible. You'll also be fine if you hit the fire at the end of your roll. If you're in the middle of a roll however you will fall in the lava.
You can also get to the other side of the second flame jet by L-walking into it with Iron Boots on at an imprecise angle.
Alternatively, instead of taking the path towards the first two flame jets, you can also perform a spinner bomb boost to skip them entirely. This is faster but trickier.
There are two ways to skip this Flame Jet.
After passing the second flame jet, you can Long Jump Attack to across the lava. Simply take out the Gale Boomerang, R-Target over the giant switch that unlocks the next room, and jump across while the boomerang is over the switch.
Jump onto the small ledge near the flame jet, and wait for the enemy in the lava to spit a ball of lava at you. When it hits, run until you're at full speed then quickly roll through the flame jet.
Quickly roll to the right (GCN) to land on the platform below and grab the key, then quickly make your way to the rotating platforms. If done fast enough, should be able to make your way across the platforms without having to wait for them.
In the third room of the dungeon, instead of pulling the stone slab out all the way, pull it back three times, then quickly walk around to the right side (GCN) of the slab and roll into it to clip inside of the slab. Keep rolling until your pushed out on the other side of the wall, and make your way to the door.
Using moonboots (force unequipping iron boots on a specific frame of a jumpstrike) it's possible to get extra height from a jumpstrike. Doing this with a precise bomb boost it's possible to get up onto the second floor without speaking to the first Elder, thus skipping the first key shard. Currently only useful in IL speedruns.
Whenever walking into a Magnetic pull with the iron boots, if you open up the item screen and put an item over the iron boots so that they're no longer a D pad item as you're being drawn toward the magnetic wall, you'll walk at normal speed. This saves a lot of time when walking across the magnetic ceiling "maze".
Grab the underwater key in this room, then swim over to the underwater block. Instead of pushing the block, simply dive under the fench with the iron boots and swim up to clip past it.
You can skip activating the second Magnetic pull by aiming a jump attack towards the blue switch, opening the timed door to the last section of the room
Another method to skip using the boots to hit the crystal switch is to LJA over to it by using the pipe, this is less risky but slower
Finally for 100% you could also leave the Small Key and try and collect it during the CS for the gate sliding closed. It is tight on timing
On the Gamecube, you can skip cutting the ropes and dropping the bridge with Long Jump Attack. Walk past the Beamos Statues to the end of the platform. Walk over to the right facing the bridge and position yourself so that the left half of Link is lined up with the small opening. Take out the Gale Boomerang, try to R-Target between the gap, above the door frame, and Long Jump Attack. The speed of the attack will allow you to clip through the tiny gap as long as your sword doesn't bounce off the wall.
In the room before the second key shard, you can Long Jump Attack over to the door leading to the Dangoro fight. Simply run past the first spinning platform, turn right, R-Target over the top of the door frame with the Gale Boomerang and Long Jump Attack.
Since you can't use a boomerang Long Jump Attack on the Wii version, you will have to instead use the spinning platform to skip the key shard. Get on the platform and get close to the end by Gor Ebizo's room and wait for the platform to begin to rotate. Hold Z and walk sideways towards the top of the platform as is rotates towards you. Once it has rotated a full 90 degrees and you are standing on top of it, roll towards the upper ledge to grab it.
Backup setup by Anorakkis
Using a bit of precise movement on ledges, you can reach the upper area of the room blocked by the Beamos without killing it. It is unknown how this compares to other methods for the room.
After activating the spinning magnet, quickly run into the Magnetic pull and equip the Iron Boots. Facing east (GCN), walk towards the corner on the left (GCN). When the Magnet reaches the other side, turn south, unequip the iron boots (which will flip you so that you face north), and jump attack over to the platform. Simply walk around the bridge.
Alternatively, you can LJA across without having to activate the Magnet
When you enter the room, it's easier to shoot the bridge down first so that you can run past the enemies, instead of walking down the path and being forced to kill the archers.