Thought I'd give the answers; many of these were correctly answered by Garnetpalmetto, with some help by GDB, Vikkyvik, and Vulindlela.
1. What is the name of the bar where Alex and his droogies hang out and get hyped up for some ultraviolence?
The Korova Milkbar
2. What is the name of the other bar where they nurse their wounds following Alex's attack on them while walking by the water?
The Duke of New York
3. What model car do they steal and play "hogs of the road"?
4. What is the license plate number (sorry, forgot what you Brits call it)?
DAV485Q - there must be some significance to this (?). Any thoughts?
5. What three "bands" does the brunette "popsicle girl" ask Alex if he is looking for at the record store?
Johnny Zhivago, The Googly Gogols, The Heaven 17s.
6. What shameless self-promotion does Kubrick include in the record store scene?
2001: A Space Odyssey Soundtrack on display at counter.
7. What is the combination to the lock on Alex's bedroom door?
8. How many times does Alex use the word "welly" in addressing his droogs following the threesome bedroom scene?
Six. I love saying welly, welly, welly, welly, welly, welly well.
9. What artists painting is used in the film when Alex kills the "cat lady" (the moment he hits her with the phallus)?
10. Alex refers to Billy Boy as: ...thou _____ bottle of _____, _____ _____ _____.
Come and get one in the yarbles, if you've got any yarbles, thou globby bottle of cheap, stinking chip oil!
Extra Credit: What is Alex's prison number in the film? What is it in the book? Film: 655321; book: 6655321.
I was cured alright!
"The deeper you go, the higher you fly. The higher you fly, the deeper you go."