1. What was Scott Summers' seldom-used nickname? (Not Cyclops)
Scott Summers' nickname was "Slim" and if you vaporized your food every time you took off your glasses to eat, you'd be slim too.
2. What historical effort were the parents of Professor X involved in?
Professor X seems to be implying that his parents' work on the Manhattan Project resulted in his mutation, but this seems unlikely. According to Wikipedia, the Manhattan Project grew out of an earlier effort that began in 1939. This would mean that Professor X is, at most, 24 years old in September 1963. But he looks much older.
3. Which member of the X-Men had the same name (but different abilities) as a Golden Age Marvel character?
The Angel was a Golden Age Marvel character with no superpowers.
4. Name the FBI agent who appeared in several early X-Men stories.
5. Who was the first person to decline admission to the X-Men?
CJGeers got them all right. Lito S. got all but #5. Michael Rebain got the first three, and Kryp44 chimed in with the correct answer to #5.