According to Google, the information from Baseline will be anonymous, and its use will be limited to medical and health purposes. Data won't be shared with insurance companies.

Given the revelations from Snowden I see no reason to trust Google or any other large American company.

Those rules were written in 2010 — and AT&T has pledged to abide by them for three years if its DirecTV purchase goes through — but were knocked down by a federal court in January.

Even if AT&T did abide by these rules in three years their commitment is over and they can do what they want.

To be fair the idea is for results that are libelous or potentially (legally) damaging to a person to be removed. Making someone look like a twat is not to be removed (or at least should not be deleted). Google seems to be pushing the envelope on what they are removing to provoke resistance to the law in the hope of getting it changed.

Is that the only measure of success?

I guess it is "influential" in the same way that Google news shows the most "influential" sites covering a significant story. Influential means "conforms most to the prevailing viewpoint" (at least in this case). These are the ones that the most people read and hence the most influential.

This is not claimed by the article. While Google was the one sued the article says that the ruling applies to any search provided who receives a complaint.