I don't believe the Cootie is fast than a straight key. The Cootie key was invented as an alternative to the straight key and to give some relief from "glass arm" or carpal tunnel syndrome. It became obsolete when the "bug" was invented.
The Cootie key is keyed back and forth; right, left, right, left, with whatever element is next being sent on the next left or right alternation. For example CQ is sent right (dah), left (dit), right (dah), left (dit),...right (dah), left (dah), right (dit), left (dah).
The original was the Bunnell Double Speed
Key. You can send a dit or dah with each movement in either direction. Some operators go to it because they can key faster than with the traditional up-and-down telegraph key and it's considered a straight key. The next faster straight key is a bug.
Yeah, it was considered obsolete but: http://www.sideswipernet.org/
as DJ1YFK listed above. There is quite a following.