Hey, it seems fun to try to answer. Let me try but you'll have to forgive my bad English. Also I'll try from my head only so probably a lot will be wrong.
1. A vocaloid is one of the multiple artificial voices that are able to sing that the market software VOCALOID has to offer. Basically they're like Virtual Studio Technology instruments (VSTi) that imitates human voice. The VOCALOID software is the software that lets manage those kind of VSTi easier.
2. Their main use is as artificial singers, be it directed to a professional market (there are a lot or VOCALOID albums that can be bought) or for personal entertaiment.
Of course, being artificial voices they can be used for simple talk, too. But normally with worse results that a real talking software can produce.
3. You buy each "voice" standalone as they include always the VOCALOID software it is needed to manipulate them. Then you install that legally bought software in your computer with a compatible OS (normally Windows only but not sure) and use the program similar to any music sequencer (asigning pitch, volume and other caracteristic for each note but also asigning the voice sample used).
4. Hmm, this one is hard... I'm sure I'm going to forget some...
LOLA, LEON, MIRIAM, KAITO and MEIKO for the original VOCALOID.
For VOCALOID2 (an advanced version of VOCALOID) I remember Sweet Ann, Big Al, Prima, Tonio, Sonika, Hatsune Miku, The Kagamine twins/mirrors (whatever... I'll count Rin and Len as one as they're in the same pack), Luka Megurine, Gakuppoid*, Gumippoid*, Gachappoid* (* probably I wrote them with some letters missing or extra), Kiyoteru, Yuuki (or is it Yuki?), MIKI, Lily and VY1 (Vizyi1 or something... or is not released yet?). A new one called iroha is almost ready to sell, too.
(Aside there are the fanmades, gender benders and even talking software from one of the companies that make vocaloids that seems a lot like vocaloids but that's not VOCALOID software.)
So... 22 released right now or almost released that I can remember.
5. I think VOCALOID will have around one to three million fans around the world -mainly in Japan- (let me make that aproximation thinking in numbers from the Vocaloid ranking... probably it's a lo~ot less). I'm not sure if that's being popular from a global point of view.
But if you think it is then I must say everything started from the idea of antromorphing one of the voices (Miku Hatsune) into one "pop star Diva" giving her an anime like appearance. NicoNicoDouga and viral video spreading did the rest.
6. What brings the question of what is NicoNicoDouga (NND). It's another portal where you can upload your videos just like youtube. But it has a pair of characteristics that are appealing to the Japanese market: anonimity and the famous scrolling comments played with the video the instant you typed them (giving a time dimension to the comments).
7. Yamaha is the owner of the VOCALOID software (other companies just used their software and release their voicebanks. Yamaha only has done VY1) so I guess some of their software of artificial instruments could be related.
AHS (a company that made some vocaloids) have talkloids (programs to make talking) that have a similar animeish vocaloid marketing.
Hmm, this question was very vague.. I don't know.
8. The same way a chainsaw and a knife are used to cut, UTAU is like VOCALOID as it's also used to make a singing human without having a real singer physically.
It's a lot different on them. The main ones I can say shortly are that UTAU is freeware (except a version that is shareware), use different way of obtaining the voicebanks (VOCALOID kind of VSTi singers are way more complex than the human voice samples of UTAU) and they have a very different way of handling the software use.
9. UTAU isn't popular at all. -_-
I doubt there are 50000 UTAU users all around the global world. Probably even less than 20000.
Any videogame company with those sales with a worlwide market would go red numbers very fast.
About fandom... the most popular UTAU video in youtube is Kasane Territory with near one million views in one year and probably a lot came from the Touhou fandom... while the Pink Panter Theme Song has over 6 times more views in only three years (I checked that.. I don't know that by memory >.>).
The UTAU fandom isn't very big but if it's any popular it comes from Teto Kasane and her cute relation with VOCALOID (being a fake diva joke, etc...).
10. UTAU just pitchs some samples up and down.
You can install UTAU or use the not needing to install zip option, unlike VOCALOID. Also unlike VOCALOID, the voicebanks -except the default one- don't come standalone and need you to download the software. But you can just use one UTAU software for any number of voicebanks so it's not very troublesome. It's also similar to a piano roll keyboard in use.
11. Known in the internet there have been over 1400 voicebanks
(I had to check this as I don't know the direction on memory) acording to this website. Of course, there could be a lot more not released in public.
12. UTAU just pitchs samples to make a sound acording to the music, then you must record enough samples to cover all the possible sounds there could be in the song (just the Japanese syllables for Japanese or add more complex voice samples for English, Spanish, etc). Also as pitching isn't perfect if you go too far from the natural position so it's also seen to make various samples for the same sound but in different singing pitchs so UTAU can choose and being more natural.
The syllable method (consonant -C- plus vowel -V-, called CV) is the normal and basic way of doing a voicebank as Japanese mainly works that way. But there are other ways as VCV or CV VC.
With these methods you can make the words phonetically with those samples.
You could just do a "word by word" voicebank just recording all the words in a dictionary and using as samples each word. But obviously this voicebank is going to be huge in size (100k wav files for a basic English?) so the normal methods are the only ones used that I know. Also, as the music aims for different notes and length each vowel, to program in UTAU a "word by word" voicebank would be very hard -if not impossible-.