Muggles' Guide to Harry Potter/Magic/Phoenix

Overview
A Phoenix is a magical bird that periodically burns up and is then reborn from its own ashes.

Extended Description
Phoenixes are truly remarkable birds. While their rebirth from their own ashes is relatively widely known among Muggles, it is not generally known that they can lift extremely heavy loads, that their tears have healing powers, or that they are extremely loyal pets. We are introduced to one phoenix in the course of the books: Fawkes is Professor Dumbledore's pet phoenix.

Analysis
While we are not introduced to the bird itself in Harry Potter and the Philosopher's Stone, there is mention of the phoenix in that book; Mr. Ollivander mentions that there is a phoenix feather at the core of Harry's wand. Ollivander mentions also that the phoenix in question has only ever given two feathers; but since he mentions that his wands are made with phoenix feathers, dragon heartstrings, or unicorn hairs, one can safely assume that there are other phoenixes that he can harvest feathers from. Note that Ollivander selecting these items for the core of his wands indicates that he believes them to be the most powerful magic items available.

Fawkes' special powers play a very large part in Harry's defeat of the monster in the Chamber of Secrets: it is his loyalty that calls Fawkes to Harry, the healing powers of his tears are what restore Harry to health after he is injured by the monster, and it is his ability to lift heavy loads that allows Harry to escape from the Chamber with Ron, Ginny, and Professor Lockhart after the monster has been destroyed.

His loyalty and intelligence are extremely useful to Dumbledore in Harry Potter and the Order of the Phoenix, where Dumbledore has Fawkes carry messages and keep a lookout.

Questions

 * 1) Why is there a phoenix feather in Harry's and Voldemort's wands? Does it signify a wand whose wizard/witch's strength is duelling?
 * 2) Phoenixes are shown as not really belonging to the realm of ordinary magic, or ordinary Magical Beasts, but quite extraordinary; and there is only one we see in the entire series. Yet for Ollivander, phoenix feather is a quite regular material to create wands from. How many phoenixes then exist?
 * 3) Most people would, it seems, wish for a Phoenix as a pet - but we cannot imagine them being sold in the Magical Menagerie in Diagon Alley or some such place. How then do you get a phoenix? What do you need to be to "qualify" for having one? How did Dumbledore come to own Fawkes?