The Zeta-beam only appeared in the Southern Hemisphere; the idea was that it only appeared on the closest half to Earth, and that was (spatially) the area south of the Equator.
2. With what museum was Adam Strange associated?
Adam Strange was associated with the Metropolitan Museum as we found out in Mystery in Space #87:
Incidentally, this story established that Adam's museum was located in New York City; there is no doubt about that as Carter and Shiera are mentioned as driving, "along a highway entering New York City," and the couple also attend a Broadway play during their visit to the Big Apple.
3. Who was Adam's girlfriend?
Adam's girlfriend was Alanna.
4. Who was Adam's girlfriend's father modeled after?
Alanna's father Sardath was modeled after Julius Schwartz. From Schwartz's wonderful and entertaining autobiography, Man of Two Worlds (pg 85):
"... incidentally, Sardath, the father of Alanna, was based on an Infantino characterization of me."
5. What villain picked up amazing mental powers from a rock Adam brought back from Rann?
Ira Quimby, aka, IQ, shown with the purple suit and blue bowtie at the extreme right of the panel shown in the answer to #2 above. Quimby gained mental abilities from a rock stored at the Metropolitan Museum that enabled him to plan elaborate robberies with amazing gadgets. However, he was a Hawkman villain, not an Adam Strange character except as to his origin.
Jim got 1, 3 and 5 correct, and later identified 2. Ian correctly identified 4 as Julius Schwartz and later chipped in the answer to 2. Robert McKinney got 1, 3, 4 and 5 right. Michael Rebain answered 1, 3 and 4.