I'm starting a minimal jumping run in medium mode and putting up my results. If you think any segments could be improved, or want to know how I achieved one of them, post here. I'm starting with the upper path (complete! I'm starting on the Kraidzeif path now), and counting regular jumps, midair jumps, wall jumps, pretty much how many times I press Z. If a save point requires an extra jump to get to, I can't use it.
Start to 1st medium save:
12, and the save is at the topmost left block
There, skipping the next save, to the next wuss save:
9There to just before Mike Tyson:
10I'll skip the bosses until the very end. They're going to be h-a-r-d to get a minimal jump count on. So:
save after Mike Tyson, skipping Legend of Zelda save, to save at the beginning of Ghosts and Goblins (block at far left):
31There to wuss save after spike corridor:
10There to next save:
13 but don't even ask about bat-riding the whole way
There, skipping the next wuss save, to the wuss save after that (block on far left):
11There to the save before Mecha Birdo:
11Again, I'm skipping bosses until the end...
Save after the save after birdo to the next wuss save:
12There, skipping fan room save, to the wuss save at the top:
9, and I save so that I can shoot the on/off switch on the way down
There to start of Castlevania:
9, saving on the second step from the top, to the next wuss save (I jumped at the same time I save so that if I held left when I spawned, I landed on the one-block-wide platform to the left of the wuss save):
17There to the save before Dracula:
4Again, skipping bosses...
upper path complete! Total: 158 I think.Room of Divine Transportation (block on the right, even with the save) to next save (just to the left of it):
28. FYI, 13 before tetris, 12 in tetris, and 3 after tetris.
There to the save in Kraidzeif's hideout:
31There to next wuss save:
8There to save before Kraidzeif:
14There, skipping the save at the beginning of Megaman, to the save where you can decide to go down or to the right:
11I'll take the Bowser path first.
There to wuss save:
7Skipping bosses, but for the record just the boss is
4.
Save in the "oh sweet god" room to next save:
13As we've already seen, there to wuss save at the top of fan room:
21There to beginning:
4
Then on my subsequent medium run-through it took me upwards of 100 deaths to get it. Ah well, better that than the other way around.
On the walljumping segment after the tetris room, you need to do some quite tricky jumps to get through it minimally. So. It took me around 200 tries to get through the first time, and I knew how hard the next segment was going to be, so I... saved. 
I'm sorry guys, if you hate me that's fine but first do the series of jumps attached in the screenshot. Then do the next segment.
