The falling spike balls in the beginning can be avoided by throwing very well timed egg shots to break the cracked blocks that the spike balls are resting, and then quickly running underneath.
The door with the key in it can be accessed by several methods. The koopa shell method is shown in the video, but the arrow lift can also be tongued through the wall as shown in the gif.
You can perform a gatehack as a backup strat incase you fall down without any way to reach the door on the right, simply enter the door on the left and make your way to the gate.
It is 112 frames (1.86s) faster if you're in the middle of the screen rather than the far left side before the boss fight starts. This is because the screen doesn't have to move as far when the blocks explode. This can be seen via the difference in the frame count in each of these images:
Get multiple eggs from Hookbilll at once: 
The blast-off trick on the boss is performed by ground-pounding on the boss before he lands on his back for the final hit. This can actually be detrimental to perform during a run, because if done wrong, the game can freeze for several seconds to up to a minute at the keyhole screen.