1. Does the Dot communicate directly with one's smart phone for anything (specifically to listen to Amazon/Audible audio or Kindle books) or is this all done via the internet? I'm hoping I can use my phone in San Diego for any or maybe some of Dot's functions requiring one while mom uses the Dot in LA.
If I'm understanding your question correctly...you can use your phone (or other mobile device running the Alexa app) to access your Alexa account from anywhere from which you can access the internet, and from there you can control/configure your Echos, Dots, etc in several ways. One of the things you can do from your mobile app is select music or other audio from streaming services you have integrated with your Alexa account (Kindle books, Pandora, iHeart Radio, etc) and have it streamed to whichever of your Alexa-enabled devices (Echo or Dot) that you wish.
2. The Dot should arrive any day...I hope to set it up and test it and then give it to her to use. Can the unit can be reset to listen to her voice (or anything else that my testing it may 'set' on the unit or service)?
While you can (and should) engage in the speech recognition training function provided by Amazon, you do not need to train/retrain your Dot to understand any single individual. It will understand anyone who speaks clearly and at a reasonable pace.
3. Would my mom need a smart phone with her for any other of the Dot's functions?
Only if she wanted/needed to change any of its configuration options, use the shopping list function, review past voice commands she issued or any of a handful of other minor operations that she's unlikely to require.
4. She loves to ask questions...Dot does this but how well? Can it be set to use Google (or particular search engines) to answer questions?
No, you cannot select a specific search engine and, as you might imagine given that Amazon and Google are competitors in this space, Google is Alexa's go-to information source. Exactly what it uses depends on the nature of the question. Some things access Microsoft's Bing, others invoke Wikipedia...while still others (like questions about local movie screenings, restaurants, etc) will invoke other online services. As for the quality of general random questions, that is an area where Alexa suffers in comparison with something like Google Home...but not so much as to seriously handicap Alexa, especially if you're mostly looking for useful information rather than idle trivia.
5. Is there anyway that she can say "call <name>" or "Call 911" and it will function as a speaker phone?
No.
6. The perfect thing would be for her to use it to check out audio books from the library...any way to do this (without having a smartphone which she cannot see to use or to charge)?
That depends on what you mean by "the library".
Can it be used to listen to radio stations?
Yes, provided the stations you want are available via one of the streaming services with which Alexa is integrated.
8. Can it be used to get local tv station listings (although she is almost completely blind, we got her a big screen tv she likes to watch)?
Yes, via optional 3rd party supplied "skills" that you can selectively enable.
9. Back again to audio books...can the Dot list the titles in her cloud (again without a smart phone at her location)?
That I don't know, as I've never tried it (as it would be terribly cumbersome compared with looking something up via the app).
10. I saw a friend's Dot and the documentation that comes with it...not a lot. Is there a link to get further info?
Have you tried a Google search on "Amazon", "Alexa" and "Dot"? There is a wealth of info available on the net about these devices and the Alexa service, but from Amazon and other sources.
11. Echo could improve her life so much (and anyone in her position)...can functionality be added to it?
Yes, via the aforementioned 3rd party "skills" (though you need to have some software development abilities to do this yourself, unless your needs can be met by one of the already existing skills, which are constantly being added to).
Finally, there are so many elderly people who like my mother, is deaf (but can still hear using headphones), blind and bedridden. Something like Echo could open up the world for them and I would certainly be interested in assisting any effort to make Echo functional enough to do so and a way to provide it to them.
Since the Dot has the ability to send its audio output to either a wired or Bluetooth external devices, headphones would certainly be useful for her.