With the boots and hookshot, the fastest way to the north door is to bonk off of the pot after hookshotting to the block. Doing so requires you to first move up or right and then dashturn, to avoid grabbing the block. The difference is minimal, but generally moving up before then dashturn will save a frame over moving to the right.
It's technically also possible to hookshot at a precise spacing such that there's enough space between Link and the block to avoid grabbing the block, allowing you to dashturn without moving first, but this is frame-perfect and not recommended unless you can consistently react to the pixel position of the hookshot.
Lifting the pot and walking around will allow you to collect the small magic refill, at the cost of roughly half a second. This can be required if the boots are not available.
With boots but no hookshot, two bomb jump setups are available to get to the north door.
The first involves multiple gap skips using bonks -- for these you will need to hold in the direction of the bonk to escape the pit slip zone after landing. The final bomb jump is a standard northward bomb jump, where Link can be positioned at y-coordinate 1070 or 1071 when the bomb explodes.
The second, faster route, involves a bomb jump lineup to make it across the first gap. If using a spin slash setup, it will be faster to perform the vertical lineup first to avoid having to re-slash after the spin.
This animation shows the 4 possible positions that the bomb can be placed:
Hovering across the initial gap will save time, though this depends heavily on your consistency with the hover.
Without the boots, two separate bomb jumps must be used.
Because the room to the north is on a different supertile, the spike cycle resets upon re-entering the room from the north.
The two rooms on the east are on the same supertile, so the spike cycle offset will be saved if you enter those room and come back, but this is generally irrelevant.
When taking the left-side route down, you can hookshot to the block just off-screen, saving over half a second over doing an extra hook. Make sure to step left far enough to be completely clear of the middle block, otherwise the hookshot will catch on it, causing you to fall.
Coming from the north door, the left-hand route is the slowest but most straightforward.
This route can potentially save over half a second over the left-hand route by hookshotting to the right instead of bonking across the gap. Be careful to step far enough left for the final hookshot, to avoid accidentally hitting the crystal switch instead of the railing.
Optimally, you'll make it in time to walk underneath the spike for the last hookshot, but this can be tight and depends on clean execution. Being hit by the spike can potentially cost an entire 7 seconds, so a safer approach is to walk above the spike and hookshot from there instead. As shown by the timings, this still saves time over the left-hand route.
The right-hand route is the fastest and can be performed in multiple ways, though all of them are somewhat tricky to execute.
The strategy shown here uses a hookdash to cross the second gap using a spacewalk slide. Because you must be grid-aligned in order to successfully perform the slide, you'll want to ensure that you're flush against the block on the left first. As shown above, this can be done either by holding your sword out to the right and moving left, or by moving left and then taking a step upward before doing a quick dashturn.
If you didn't already grab it, you can collect the small magic underneath the pot on the way at the cost of roughly ~0.75 seconds, but this is not recommended as it's generally faster to collect this during the initial trip to DMs room (Stalfos Room) as mentioned above.
An even faster alternative is to use a spacewalk buffered hover to chain hookshot usages by performing a hookdash for the second hookshot and then holding down and A to buffer a single hover input. Unlike the spacewalk slide technique, this doesn't require you to be grid-aligned for the hookshot, but it does require a frame-perfect input: you must press Y exactly 2 frames after releasing A for the buffered hover to avoid falling.
Although slower (and potentially less consistent) than the spacewalk slide strategy, you can also use a dashturn and a bonk to cross the second gap. Ideally you'll usually want to position Link with a slight upward offset after the initial hookshot to prepare for the dashturn; doing so will cause Link to "nudge" against the block during the bonk, which will decrease the bonk distance and put him in position to immediately hookshot downward. Moving too far upward, however, will simply prevent you from getting across the gap.
Unfortunately, because of the way the hookshot works, Link's y-coordinate after hookshotting will not be consistent, and can range from 1075 to 1078.
You can get from the north door to the south end of the room with only the boots through this sequence of bonks. Be sure to hold down-left when landing from the 2nd bonk and down when landing from the 4th bonk to avoid slipping into the pits.
The same traversal can also be made via this series of 5 bomb jumps.
Link's x-coordinate needs to be 1688 or 1689 when the bomb explodes. This can be done via a spin slash setup, but a quicker method is to abuse the lower-left pit corner as shown above. When Link's y-coordinate is at 10C9 or below (shadow covering at least one row of the bright floor triangle pixels), moving into the left-side slip zone will cause Link to be pushed diagonally down-right until his x-coordinate is 1688, setting up the horizontal coordinate. Be careful that Link does not go too far downwards, as this will cause you to slip off the bottom pit edge instead.
The simplest way to execute the bomb jump is to place the bomb when Link's y-coordinate is between 10B4 and 10B9 -- these two positions are shown in the animation above.
It's also possible to move Link upwards after placing the bomb, to allow for different setups -- for example, moving Link 8 pixels north after placing the bomb will result in a completely horizontal bomb jump -- but this is more error-prone and not recommended.
If using a spin slash setup for the final bomb jump, extreme care must be taken to time the charge and spin to avoid hitting the moving spike, which will cause Link to be knocked away and fall into the pit, undoing all of the progress thus far. A slower but safer alternative is to charge a spin slash facing upwards, then move down and release the spin, before laying the bomb.
This path through the room is used for GT basement routes which mirror to the entrance after collecting the four chests in DMs room (Stalfos Room) and then come through this room a second time, as well as in vanilla speedruns where those chests are skipped altogether.
With clean execution you can hookshot to the lower block while the moving spike is on the far right. You'll want to ensure that you hookshot to the very edges of the two blocks to make this possible.
Alternatively, using slower movement through the first part of the room will allow the spike to pass back to the left, leaving a clear path to hookshot to the block at the cost of around 20 frames.
These bonk/bomb jump setups can be used to reach the second half of the room, at which point you can proceed with the other strats shown above in Without Hookshot (2) or Bombs Only (2).
Since the door next to the crystal switch is a "Bastard Door", holding diagonally up-left in the doorway will lose 2 frames compared to using a cardinal input. If using the cardinal movement, note that there is only a ~3-frame window for getting optimal cardinal movement but still transitioning to diagonal early enough to avoid getting stuck on the switch.
From Item Use Speed, we know that hitting the crystal switch the first time is faster with the hookshot than with the boomerang or a slash. The main advantage for using a slash is that it may potentially help with the Stalfos if it's in the way.
One of these three strats is generally used for hitting the switch the second time while crossing the pegs. The boomerang strats are the fastest, and essentially equivalent in time, with somaria being slightly slower.
The somaria dash is optional but ensures that there is no delay between exploding the somaria block and dashing. It costs 10 frames in the next room.
As a last resort, a bomb can be used if none of the other items are available, though this is clearly slower than the other strats.
If the Stalfos is not in the way, hookshoting to the pots after hitting the switch can save ~6 frames over walking.
Dashing out of the doorway from the pegs can potentially save ~4 frames over walking, but requires clean movement and execution. Link's vertical coordinate needs to be at 1194 or below in order to avoid bonking into the pegs.
With the red boomerang, one of these two strategies can be used, depending on both your execution as well as whether the Stalfos is in the way:
Throwing the red boomerang to the right is also possible, but not recommended as the execution is extremely precise: