OK, some new analysis of this game and a progress update.
Aerial speed, either from jumping, high jumping, double/spin jumping, or falling, is slower than running along the ground, so max movement time should take place in a run attached to the ground. There's a delay when jumping of a few frames, and a slight delay when beginning the double/spin jump. If a long or spin jump can help avoid doing two regular jumps, the additional time in the air appears (I haven't measured) to more than make up for the delay of launching an additional jump and spending two regular jumps in the air. For example, jumping a shuriken with a regular jump might be fine, but if I also have to jump the next guy right after him I would rather try extending my air time with a high or spin jump and see if I can get the second (or even third) guy to throw with poor aim, negating the need to do a second jump altogether.
I did some playing around and I'm not 100% certain what determines whether you stop or keep running when you hit the ground after a jump. I thought I had it but I was wrong. In my first video I managed to use air attacks to hit two samurai and keep on running. I was not able to finagle that same trick in other areas of the level against ninjas when simply playing around.
In the second video I do much better jumping and path selection, because I lose over 300 frames when I stop to get the sword power up (you stop running and there's a screen-filling animation), yet I'm only behind a net 50 or so frames going into the boss. I kill the boss almost 400 frames faster with the sword power up, making the 300+ frame loss from getting the sword worth it. There is a little awkwardness when getting the sword that I'm not sure how to deal with, but I may see if I can shorten the delay in the lead up to the powerup.
Here's the second movie file. Note that it has the same name, so rename your original.