>Maybe a movie clip is a copyright violation and maybe it is fair use. Does Twitter take down that content as soon as there is a complaint? Does it have to wait for some legal action to start or finish? What is the recourse if something is taken down and later found to be legal content?

We have the DMCA to cover this... however it's currently an extremely broken process. While there is a penalty codified for making claim to something you don't own, it is never enforced. That part of it needs to change and the regulation should then act in the manner it was designed to.