How is the letter "R" pronounced in French?
Or rather, how is it produced in the throat? Please describe this in as much detail as possible. How come some roll their r's and others sounds lighter and without rolls? (ie: Rrrrr) What's the difference and is there a preference? Are there culturally based factors like which region or country a person is from? And lastly, how can a throat rrrrrrr be sustained and alter in pitch such as how Edith Piaf sings her rrrrrrs?

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The rolled "R" in french is not a basis, there is no fixed rule about it. Some roll the R saying that that gives it greater linguistic authenticity ( as the R in Latin language is rolled ), some pronounce it just like in english.
It depends on regions and dialect. But as long as the meaning doesn't change, this is not a big deal.
I remember  that my teacher told us in a phonetic lesson a funny story about that, one would say that the french king Louis 14 could not pronounce the letter R properly ( rhotacistic ) that's why linguists at that time made the rolled R a rule :)

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You could try this video: http://www.youtube.com/watch?v=NQA4Wyhb8BU or other videos on youtube.

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The pronunciation can depend on regional differences. In most of France the 'r' is a purr in the throat, as you say. In some accents of southwest France, though, the 'r' is rolled, more like in Spanish.

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In french, "R" is pronounced silently... And you need to put just a tiny bit of "h" sound at the end of "R"

Ex. Bonjour (bonjourh)