If someone who owned a share of Search Google managed to get a controlling interest in Ad Google and was operating it to benefit Search Google to the detriment of Ad Google's minority shareholders, that would be one of the rare cases where a court would intervene.
That's fine, but "fiduciary duty" is a specific legal concept and you are misusing it. Making shareholders angry enough to vote you off the board is not the same thing.
Also, it's worth noting that institutional investors often rubber-stamp the board's recommendations in shareholder votes. One side effect of the rise of passive funds is that there is less shareholder opposition to board moves nowadays.
No, they don't. In fact, I can't find any that involve maximizing anything.
Also, filing a lawsuit is trivial. Succeeding is a whole other kettle of fish.
That's exactly the myth the above link is about, btw.