I am not sure what you are looking for, but here goes.
It sounds like it comes down to content removal, what it takes to have content removed and somehow does court ordered removals play a bigger role in having Google cooperate, then the answer is NO.
Most reputable or ethical online businesses of all sorts especially the Googles, Yahoos etc. have a way for anybody to request removal content for a variety of reasons based on infringement.
It is called the The Digital Millennium Copyright Act (DMCA) is a United States copyright law that implements two 1996 treaties of the World Intellectual Property Organization (WIPO).
What's left is most likely some other form of either illegal activity, harassment or privacy matters, and those can be reported in most cases and the company will act.
You do have to file each report at least 2 times. One to the website hosting the content and one for Google. Each complaint/report must contain a Legal contact info, seperate infringement and removal request.
In the end, does the Court have the ability to order removals, sure but so can you and I and we are taken just as serious, and I have filed for copyright infringements online about 50 times.
What a court can do is subpoena records, which can do through the free information act, and all other matters, well... through the court.