Finally redid Stage 2 and 3, saving a total of 573 frames over the last version I posted. Here are the improvements:
Nothing changed in stage 1.
Stage 2
Finally one of my ideas worked, that first vine I grab can be used to go through the floor, but only if you have the max speed from the vine, and only where the slope meets the horizontal platform. This didn't save as much time as I hoped because unfortunately the animation from falling too far can't be avoided. Still saved 69 frames from this alone.
I didn't think the next vine could be as useful, but after a rough test I did the other day I found out that you can preserve the momentum enough for that little slope to give you the max boost. Doing this is actually faster than walking slowly to the right, but getting that next slope to recognize rex was really difficult. Plus it looks a lot cooler.
The next vine done in 4 swings like you suggested, and it saved around 30-40 frames but I didn't check the difference exactly. I also used the 4 swing on the momentum boost stated previously, which only loses like 60 subspeed, but overall way faster. Thanks for bringing the 4 cycle swing to my attention: it turned out to be pretty useful so far.
Then various very minor improvements here and there.
Stage 3
This stage definitely looks different than before, but I'll just point out the most obvious changes.
Bypassed bouncing back and forth in one section when coming out of one of the tubes by slowing my momentum just slightly.
Avoided a top route in favor of a bottom route because of the pesky birds. Plus I can get a nice momentum boost on the lower route without those things that slow you down. So there won't be any damage taken here anymore.
Near the end where I previously took damage on that slope, I found that you can still be pushed forward while roaring, so timing that just right will clear the green thing that's in my way and allow me to continue moving without wasting time taking damage or killing it otherwise. I'm quite happy with this find to be honest.
Without further ado:
http://tasvideos.org/userfiles/info/19610312690352114
Edit: Okay, 2 of the flame powerups in level 9 are completely way out of the way compared to easy mode. One of them you have to kill one of those armored dinosaurs with level 1 flame to be able to jump to it, after you take a long detour... and then the other one is a shorter but still kinda long detour. I think I'll have to get them so that I can hit both openings on the boss. I got to the boss with level 3 flame playing in real time and it did 1 damage per hit. That just won't cut it, so long detour it seems it is...
That being said this is still a vault game but figured I'd post it anyway.
On another note, I've used the double flame to bypass a few enemies faster by setting it up in the water like you suggested.
Also want to point out that it's possible to get on top of the map above the boss by freezing one of the frogs and jumping over the top of the screen, but nothing interesting can be done from up there except jump off to the right and die:
But at any rate, I improved the boss by setting up the triple flame in the water. Maybe it's possible to setup a quadruple flame, but that would cost time because I'd actually have to wait to take damage a third time underwater, but it might possibly save a few frames. I'll give it a go before moving on but I'm not convinced it will be worth it.
On the last skateboard section, jumping off onto the mushroom is actually slightly faster by about 7 frames compared to just riding the skateboard normally.
I saved 70 or so frames from the level alone and 319 total including the boss:
http://tasvideos.org/userfiles/info/20971140300926996