See video for demonstrations.

Level 5, combining double jump with flame jump is human viable and probably possible to add to runs. Copy of TAS strategy

Level 8, bonking your head on the ceiling after a double jump may (Or may not?) be faster than jumping to the left. Requires further testing.

Level 13: Faster than TAS strategy by approx 660 ms

Level 14: Human viable version of TAS strategy

Good catch on level 13. ("How did I miss that?!")

Level 5 alternative strat. I'm pretty sure this is faster. Confirmed possible by Xanth

Wow, I tested that for hours, truly amazing that it is confirmed possible. :)