I suppose it's only fair to publish the answers. Here they are. And, of course, if you haven't seen the movie, this contains spoilers...
The questions answered:
- Why did Rick tell Capt. Renault he came to Casablanca?
For his health (also acceptable: for the waters)
- What did Rick tell Maj. Strasser when the latter asked his nationality?
- What color did Ilsa wear that day in Paris?
- Who wore grey?
- What about Ugarte did Rick object?
That he was a cut-rate parasite
- Why was Capt. Renault unconcerned about the gun pointed at his heart?
It's his least vulnerable spot
- Who is Ferrari going to pay for the cigarettes?
- What were Capt. Renault's instructions following the killing of a character at the end of the film?
Round up the usual suspects
- What did Capt. Renault say he was shocked (shocked!) to discover happening at Rick's?
- What kind of man is Capt. Renault?
He's just like any other man, only more so
- What watch?
- How much did Ilsa offer Rick for his thoughts?
- If it's December 1941 in Casablanca, what time is it in New York?
My watch stopped
- Where did Rick hide the letters of transit?
In Sam's piano
- Which character said, "Play it again, Sam?"
Trick question. No character in Casablanca said those exact words. Ilsa asked Sam, "Play it, Sam. Play 'As Time Goes By.'" Rick said, "You played it for her, you can play it for me! ... If she can stand it, I can! Play it!"