I believe stopping while in midair makes you fall faster than if you're moving and falling. The sooner you land on a platform, the sooner you can jump again, so try to make jumps as short as necessary. These little timesaves add up throughout the run.

In level 5, I'm not sure if it's possible to get a faster cycle on the higher moving platform because of a weird quirk: the 2 rightmost blocks on the platform with the enemy have no collision, and for some reason when your attack passes over them, it disappears. So you can't quickly kill the enemy and jump up to try and make the cycle. I theorize it could be possible to attack the enemy from the left while making your way over, but I haven't tried it.

Dying on a moving platform causes a glitch in the player state where the game thinks you're on a moving platform when you're on the ground and vice versa. When you respawn, you automatically move left/right or up/down as if you were on a platform. It makes completing the level a bit harder. To avoid this, don't die.

At the end of level 10, I'm 90% sure you can't jump to the 3rd moving platform until the 3rd cycle. It really looks like you can make it on the 2nd cycle, but I've never gotten it. If you try it and fail, you have time to jump back into position for the usual cycle, so I sometimes go for it just in case. If it's possible, it has to be something ridiculous like a 1-frame window.