By using a death with a bottled fairy, it is possible to reach the switch on the first floor that opens the door to the staircase room, skipping going around the first floor, including the first Lizalfos fight.
It is also possible to do a recoil jump and grab the ledge to get to the switch, which is a much better option as it only requires a Deku Shield and not a bottle.
It is possible to cross the flame circle plaforms without activating the eye switches by landing at the corners of the platforms using either a sidehop and jumpslash, or a angled jump.
If you are swordless by either having a NG+ item on B or if you just did not get the Kokiri sword, you will need to defeat the Lizalfos with deku sticks. However, it takes three jumpslashes to defeat them. So here you need to perform the Broken Deku Stick glitch by jumpslashing a Lizalfos up close. It is easiest if you stun one of them with a deku nut and then jumpslash.
Using a precise hookshot jump, it is possible to reach King Dodongo's room without lighting the eyes and going through the back area of the dungeon.