Angler Key Skip

It is possible to get into Dungeon 4 without the Angler Key by using a Tracy Boost or a Like Like Launch to get onto the Angler Keyhole and then walk off the ledge into the waterfall. The entire lower section of the waterfall does not have any collision, meaning you are free to move as if it wasn't there. This skips the Marin beach date, the Walrus cutscene, and the Lanmola fight.

Tracy Boost method

You need to have Tracy's Medicine.
Bring a Fox from Mabe Village to the ladder just east of it, place the Fox as close to the ladder as you can while still being able to climb it, talk to the Fox, immediately place a bomb and climb to the top of the ladder to get the Fox to jump up the ladder.
Bring the Fox all the way to the ladder that leads to the Angler Keyhole in Tal Tal Heights, push it against a corner of the ladder, slash the Fox, immediately push it a little bit, stop and shield before it attacks you. This should have the Fox bounce down the ladder.
Bring the Fox to the Angler Keyhole and perform the Tracy Boost by having Link clipped inside the Fox and jumping while pushing against the Keyhole.

Like Like Launch method

Bring a Cucco from Mabe Village just to the right of the Angler Keyhole. Optionally but recommended, powder the Zirro because it can kill the Like Like and interfere with its path.
Bait the Like Like to the East of the Tal Tal Heights Warp in the narrow path leading towards the Keyhole, if the Like Like isn't perfectly lined up you can line it up by having yourself eaten while you are standing in the path.
Place a bomb on the right of the Like Like to force it to go left (could be done with the Cucco but quite a bit slower) and throw the bomb away so it doesn't kill the Like Like.
As the Like Like gets past the Tal Tal Heights Warp, slash it up against the top wall and make sure it doesn't see you by heading right a little until it hits the wall or the Cucco (the Cucco could get in the way but you can deal with this by jumping around the Like Like and throwing the Cucco over it so it turns around again).
Once the Like Like is past the Keyhole, walk away so it is no longer on-screen (this makes it turn around) and bring the Cucco to the Keyhole so you can perform the Like Like Launch against the south wall (if you have trouble having enough time to get positioned before the Like Like gets to you, have the Like Like go a little further west before walking away, but not too far as it might end up leashing away from the keyhole).

Video guide on Speedrun.com

Waterfall Skip

If you do Angler Key Skip, you won't be able to easily make it to Eastern Tal Tal Heights and Dungeon 7 due to the waterfall blocking your path. You'll need to get around that waterfall if you want to make it to those areas.

If you're also doing Flame Skip

Use rooster to get on top of the rocks below the Western Tal Tal Mountain Range Warp just outside of flame cave.
Jump carefully to stay on top of the rocks and make your way to the falling boulders ladder to the east.
Do a noclip from the ladder to get on the mountains in the background.
Go around the Wind Fish Egg.
Use a Pegasus dash jump to get back in-bounds to the west of the hookshot bridge.

Video clip on Twitch

Obsolete but still valid option

Do a noclip from the ladder leading to the Wind Fish egg.
Navigate to the out-of-bounds mountains in the background.
Make your way to the right side of the Wind Fish Egg.
Use a pegasus dash jump to get back in-bounds on the upper area that leads to the ladder on the other side of the waterfall.