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.
Consider that Verizon and AT&T, the two largest mobile phone providers in the US, are both Baby Bells resulting from the break-up of Bell Telephone Company in an anti-trust action. They compete strongly against each other, in a way that would absolutely not be true if they were still the same company.