Kakashi's no speed demon but he does fair pretty well against thosw that are.
Examples: Kakashi matched Itachi's speed in part 2, Kakashi's raikiri blitz on Kakuzus first heart, also his near instant chidori attack as a kid when he wiped out nearly 20 kage bunshin in seconds.
I wouldn't go as far to say he copied Gai's gates speed w/ his sharingan b/c it seems pretty impossible to produce such results by copying a jutsu but I guess it is plausible since he could open one gate outta the eight. He could have used it like Goku's kaioken. He waited for the right moment to ustilize it taking advatage of it's lasting effects, possibly when he and Gai charged at the Edo Jins. Kakashi's a genuis w/ natural born talent so I don't doubt that he'd formulate some training to nullify Gai's incredible speed. Also the Eight Gates grants its user ability beyond a shinobis capabilities and limitations so it wouldn't be right to say that a shinobi as gifted as they might be could match a Gates user. Its power cannot and should not be matched unless another forbidden jutsu is revealed to have to same effects if not greater.
But nice thread reguardless, man.. I'll rep you anyway..