If the author is truly SELF-published (author owns the ISBN and the publishing rights and managed the process), probably so. But if the book is contracted for someone else's publishing, maybe not. That will depend on the terms of the contract. Not a laywer here, but that seems pretty basic.
That's not clear. Amazon has a feedback option, and an author can make changes – for example, if the publisher sent them info indicating the book is 302 pp but it’s really 320 pp, the author can notify them and Amazon will change it.
But the publisher has more leverage – they’ve already sent the book, and if Amazon hasn’t gotten around to including the look inside feature, I doubt that a request from the author will make any difference.
Best selling authors in my genre, like Jim Butcher or Kim Harrison, have the feature on almost all their books. (Although ironically, they’re not the ones who really need it) But for the rest of the UF authors it’s hit or miss. Rob Thurman who is quite successful, has the feature on the first book in her series, but not the rest. Another author I know, less successful, has his first three books all with the feature.
Some publishers seem to have more presence than others. Bantam Spectra and Eos seem to do particularly well in terms of getting it implemented.
However, it’s not worth worrying about. In my genre, at least, Amazon sales account for a surprisingly small percentage of total sales – bookstores are still king, though this might well change in the future.