The hitbox doesnt include the whole sprite, only left-middle to front, but not the cape. You should worry more about jumping early, than when you land.

You have to hit the right point when jumping up the wall, there is a checkpoint right at the start of the section so you can practice.

In the last room shot while jumping on the right side, so you can instantly enter the warppoint

You can Kill yourself by pressing Q

You can restart the Game by pressing F2