1. Who told Barry Allen, "You're my best friend, Barry!" and after that story never appeared again in the Silver Age?
Fred Pearson, an actor who played the part of a gorilla in a theatrical production. In the story (from Flash #106), Grodd is the gorilla who's menacing the city, and Fred wonders if he's to blame. Fred does not appear again.
2. What Flash villain was inspired by a competition to represent a game company?
Captain Boomerang was created by the W.W. Wiggins Game Company to advertise their new hit toy boomerangs, but the person they chose to represent the character turned out to be a crook named Digger Harkness.
3. What Flash villain had the reverse name of a famous outlaw?
The Trickster's real name was James Jesse.
4. What Flash villain passed up a lifetime in another dimension with beautiful women waiting on him hand and foot?
The Mirror Master, as discussed here.
5. What college did Barry Allen graduate from?
Barry was an alumnus of Sun City University.
Jim got #2, 3 and 4, as did Michael Rebain. Dan of the Beat Down got all five correct. Great job!