Unfortunately, the difference between classical (fusha) and modern Arabic(s) is catastrophic; the primary reason being that people in Arabic-speaking countries all speak their own local slang dialect.
Why is this a problem? Classical Arabic focuses a lot on grammar and number. It's important (and, in fact, necessary) to speak slowly, because of the complexity of applying all the rules correctly. In pretty much every slang version, they take major shortcuts and throw out grammar -- to the point that people today who speak slang struggle to understand and articulate themselves classical Arabic.
I personally have a similar background (grew up in the west, learned a lot of classical Arabic), and when I visited Makkah, I saw people understand but struggle to figure out how to reply. My advice is to stick to classical, since all the major media (books, universities, newspapers, audio lectures, etc.) are all done in classical Arabic most of the time.
As for your specific questions:
1) The difference between classical Arabic and modern slang is like the difference between Arabic and Urdu, or Arabic and English. You may find some common words and things, but overall, it's a new language.
2) There are roughly 6000-7000 unique words in the Qur'an. There are some words that are repeated often (like "Allah" which appears almost every verse) and some that are never repeated; most vary.