Day 3 and 4:
After seeking help on discord it turned out that webGL just doesn't like platform effectors. I've changed detection of collisions to raycast, and it helped. Still, getting working version on webGL is now lower priority. If it works then great, but I'm not going to waste time much to find what's wrong.
I've also started to implement mechanic of entering dream state. Currently it's working as long as you don't fall asleep while jumping. I guess it's not advised in real world too.
I didn't make much in these two days, but tomorrow I plan to fix bugs, and implement first puzzle.