I am officially
embarrassed by my previous submission. I can't even explain how I did so bad.
I am 1/4th of the way through the run now and I have saved
00:20.40 seconds. If I maintained the savings through the run the final time would be 27 minutes and 35 seconds so at very least it is almost guaranteed to be less than 28 minutes. Hell, excluding taking some damage that I didn't do in the previous attempt, I haven't even used the advice given yet and I have saved this much time so it potentially could be a low estimate.
Here is the breakdown so far.
Saved 1 frame getting the vial in the training center simply through better jumps.
Saved 359 frames in Etruria after the training center through better jumping and taking some damage but this was 90% lag reduction from smarter shooting of enemies.
Saved 2 frames on boss 1. These were through better positioning for the last shot so that I could leave the stage on the first possible frame.
Lost 1 frame fighting the ghouls. I spent a while trying to fix this but I can't. Due to the random x and y velocities while jumping I have to hold jump for an extra frame to reach the proper height to hit the first ghoul, which loses me 1 pixel moving right so I lose a frame.
Saved 6 frames crossing th broken bridge. This was due to better lag management.
Saved 65 frames crossing the river of flame. This resulted from better jumping and flying on the floating platforms which meant my x-velocity was maximized.
Finally 792 frames saved in Hinom Forest. This was due to better jumping, taking damage which saved a lot of time off of waiting, reduced a lot of lag and the luck was improved.
EDIT
So I just figured out how health worked since I hadn't died in my previous run and hadn't tested it out. After you beat each boss (possibly after you get a new item but I know it is after the bosses die) your health completely refills so I can use 1 more health boost before the first boss and 1 more before the second boss, and I have a spot where I have to wait in both of those cases so expect even more improvements.