All About Rush

Rush Damage Transfer

Shoot with Rush out (coil, jet, probably marine), switch weapons, the bullet does the damage of the weapon you switched to (like Top) even if you have no ammo in that weapon.

Rush Despawn

When watching most "top level" runs, this is most visible in Snake Man on the screen after the first Snake Midboss. 

Rush (Jet And Coil) will despawn if, while sliding, you press Start.

Despawning Rush is primarily useful to reduce lag by getting Rush off the screen. It is hypothetically useful if you need to reposition Rush as well.

All About Rush Jet

Screen Wrapping Items and Enemies

Technically not unique to Rush, this is most often experienced while riding Rush Jet or using Rush Coil.

When Mega Man goes off the top of the screen, his Y-coordinate "wraps". This makes it possible to grab items low on the screen while jumping off the top of the screen.

The easiest place to see this is in Doc Needle over the chasm. Position Mega Man on the Rush Jet over one of the large drops at the bottom of the screen, and full jump. Mega Man will collect the bottom of the screen drop.

This has a few effects.