Well, Gopher got all the ones that nobody else got, so I guess it's time to post my answers. I had fun with that last one.
1. Which movie was reviewed on "Nik's Piks"?
The movie was called "Cadets", as seen in the intro story to "Who's Who in the LSH" #7. It was a drama about life in the Legion Academy.
2. What happened to the original training complex used for the Legion Academy when they moved the Academy out to Montauk Point, Long Island?
Speaking of which, that same story issue mentioned that the old Academy building was sold to a health and fitness club.
3. What is the most-reprinted panel in Legion publishing history?
The scene of the Legion council, from ADV 316 page 6, was used as the image for the Legion Outpost letters column for many years (as seen here at the Sannings' Legion Clubhouse).
4. Twice in early Legion stories, a male Legionnaire commented that a mission was too dangerous for Saturn Girl because she's a girl. Which male Legionnaires said that?
Both times it was Brainiac 5, in ADV 309 and 319.
5. What was Mordru's original name, and who was his father?
A relatively easy one. Mordru was born Wrynn of Gemworld, son of Topaz.
6. Besides Impulse (Bart Allen) and the Tornado Twins, which Legion character(s) have a direct link to the Flash family?
The one I was looking for was Devlin O'Ryan, who was the protégé of Iris West Allen. (As an aside, I hope they'll address which Legion timeline Barry went to, how Iris got back to the 21st century when she was previously seen in the now-defunct v4 timeline, and which future Bart comes from.)
7. This could be the most obscure question I've ever asked. Which three of the following people do not belong in the list with the others, and why? In alphabetical order: Affirmative Action, Asteroid Girl, Brainiac 5, Cosmos Lad, Disgusting Boy, Elektro Boy, Kwikster, Lightning Boy, Lightning Lord, Silent Girl, Spark, and Telepathy Lad.
This one requires you to know (somehow) that I'm talking about the parody and tribute covers to Adventure 247. All of the above except for Asteroid Girl, Cosmos Lad, and Silent Girl were all in the center seat; those three were on an end. See the Legion Wiki for all of the ADV 247 inspired covers so far. Jake did come up with some alternate answers (just not the ones I was thinking of).
Special bonus holiday question: Name three stories in which we saw the Legionnaires celebrating Christmas or another holiday celebration.
The three that I had thought of were the "Christmas Star" story (which also, if I'm not mistaken, was where we saw Colossal Boy celebrating Chanukkah), the Adult Legion story where Supergirl tries to hook up Superman with Saturn Woman, and the Klordny story. Gopher found a Halloween story which has apparently totally escaped my memory.