Thanks for playing!
To pass the first spikes (vertical fall) you have to stop right on the edge and character will start to "fall" (only animation) it means that you gain fall speed and when you actually start to fall the speed will be enough to get trough spikes in time. The second set of spikes (long jump) using similar strategy. You should sprint into the wall, gain speed and then jump over to the next platform. Next simply using movable blocks.
Also near the end if you jump on the mushroom it will send you into spikes, so you need to jump over it. Originally, this area required using a bug that allows you to cancel a big jump in mid-air, but I find it quite difficult to do even if you know about it, and it could be frustrating.
And yeah. Not polished, no sound effects. Sadly, I was busy this weekend and can't spend as much time as I planned. So there is still plenty of room to experiment with these bugs.