Level 2 Timesave: A faster way to beat this level is to turn the bed ('x' or right click) and place the bed by the top chest. Its even faster to jump into the hole instead of hitting the top wall, but this is pretty much frame perfect.
Level 8: For a minor timesave, you can turn the bed and place it as close as you can to the flames on the right. Immediately after the knight hops out of bed, jump. If you placed the bed close enough to the right, you will be able to get the first chest without bonking on the wall and walking back towards the flames.
Level 9: This small strat can save 1 - 2 seconds if done correctly. After picking up the water potion, jump towards the wall to bonk off of it. The success of this jump is determined by the frame on which you jumped. If you jumped on the right frame, you will be able to jump again after getting the chest. This skips walking back against the flames.
NEW STRAT DISCOVERED BY @DeamLinX - "You need to place a bed at the corner of block next to left chest (you can only place it on the last pixel of block, or flame will kill you). After this you need to jump instantly after reaching land (frame perfect), get hit by a wall, jump through the flame and complete the level with the previous strategy." @Galliard was the first person to accomplish this "TAS only" trick (which has been proven to be humanly possible).
(From @Merl_ 's descriptions of fast tricks): Fast Level 12: It's faster to open the left chest first, then top right, then bottom right. After getting the sword, jump over the ledge to get more falling speed, and you can hit the wall and immediately double back to get to the chest. Then get the top chest, and do another jump down to hit the wall and get the bottom chest.
Level 13 Jump: After getting the sword, you have to run down some stairs. When you get to the bottom, there's a small ledge that you normally have to walk on before you hit the wall and turn around. You can hit the side of this, saving about a second. Just like in level 12, you have to jump to get more falling speed. Jump over the last two ledges in the stairs and you can do it.
Fast Cycles in Level 14 and Level 15: These two tricks involve fast mouse speed and sometimes fast resetting.
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Level 14:
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To get this fast cycle, as soon as the level screen shows up, locate the bed and either:
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Move it up to the sword or, if you've already lost about a second locating the bed, reset and then move.
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When you get to the bottom part of the level by the lower ghost, you can jump off the ledge (safer route, saves no time), or fall off the ledge (more risky, because if you were too slow getting there, you'll hit the ghost and die). I would suggest jumping, since it is safer and guarantees the fast cycle if your mouse speed was fast enough. Falling is a backup plan, if you are too nervous or something and choke the jump (this might seem it would never happen, but I've choked the jump in runs too).
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Level 15:
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Another level where mouse speed is key - locate the bed and either move the bed immediately, or reset and then move.
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You can save 0.5 - 1 s by jumping into the hole, but its safer to just fall.
@DeamLinX 's IL video - This video showcases every level done with the fast tricks. Its incredibly helpful, and I know he put a lot of time and effort into it. You should check it out!