Since so many people have been wondering, I'll give an explanation as to how I got my time that fast.
Note: to save time, I hold the proper keys for each level down BEFORE the level starts
For levels 1, 2, 11, 14, 17, 19, 20, 23, 28, 29, and 30, I hold the keys that move the elephant right and up. After the elephant jumps off the floating platform, I release up and continue holding right. After hitting the button and falling off the button platform, I pause for a second and then continue holding right until I reach the goal.
For levels 3, 12, 15, 24, 25, 26, and 27, I just hold right the whole time to reach the goal.
For level 4, I hold right and down as the level starts, and then release, re-press, and re-release and re-re-press the down button to reach the goal. I never let go of the right button.
For level 5, I hold right and up to jump on the floating platform. After that jump, I move the elephant to the rightermost part of the button. As soon as the button is pressed, I immediately tap right to fall off the platform and then start holding right again to make it to the goal.
For level 6, I do the same as Jeffitus.
For level 7, I tap right to fall all the way down to those spikes. After bouncing off them, I hit the elephant's head on the ceiling tile that juts down before falling on the rightermost part of the button and immediately tapping right as soon as the button is pressed. After releasing right, I hold it again to get to the goal.
For level 8, I start it off the same as level 1, by holding right and up to do the first jump. The "dangerous" colored tiles actually only kill you if you stand on top of them; hitting their sides won't kill you. As the elephant passes over the button, I release right for a short time in order to avoid the danger color on the edge. I then continue holding right and up to jump off the button. After releasing right again so as to not get on the platform above the goal, I hold right until the goal.
For level 9, I do the same as Jeffitus.
For level 10, I start if off the same as level 1. However, after pressing the button and falling off that platform, I don't release right and instead hold it all the way until the goal.
For level 13, I found it's more reliable to just hold the left button as the level starts. This skips the falling animation without hurting your wrists
. Other than that, make sure you completely release the left button before hitting the right button (and vice versa) otherwise it won't count and the elephant won't move.
For level 16, I start it off the same as level 1, and then just walk back and forth, off and on the button. The button has to be pressed 5 times.
For level 18, I start it off the same as level 1, but instead of releasing right, I continue holding it so as to get on the platform right above the goal. Then I let the platforms collapse under my massive weight (must've been all those donuts or something) and get to the goal.
For level 21, I hold right until I reach the rightermost part of the button. That tile will pull the elephant down to press the button. After the button is pressed, I move the elephant 2 tiles to the right in order to fall down. Once low enough, I hold right until I reach the tile 3 left of the goal. I stop to let this gravity pull me all the way down to the ground, then just hold right to finish the level.
For level 22, I use the same route as level 8 (the dangerous color level).
Overall, the level that saved me the most time was level 25 with the randomly opening door. I was lucky enough for the door to open right as the elephant approached it.