A lot of the movement tech in the published run applies to Red as well. I definitely recommend reading the submission comments: http://tasvideos.org/2442S.html
Autofiring through menus is guaranteed to lose you frames. Try to find the earliest possible frame that a menu/title screen can be cleared. It's a bit boring, yeah, but you want to make sure absolutely everything is optimized.
I'm testing on the US version (same as the published run) and it looks like starting off Earth with a slide is faster than walking into it. Fastest form of movement is to interrupt your slide with a jump, then interrupt your jump with a dive kick a couple frames later, then immediately slide again after that. It should give you the continuous, fluid movement found in the published run. You don't have to jump over enemies: If you jump just before coming into contact with them during a slide, they'll be hit and you'll continue moving, and you can dive kick > slide without losing any speed.
Apart from that, it's looking pretty good, though I'm not sure Red and Blue have enough difference to warrant two separate publications. We'll have to see where it goes, I guess. I'll upload a US version WIP later today in case you want to study that. Maybe we could collab on either this or a Blue improvement?