As always, you guys came up with some answers I never would have thought of, especially with my first question.
1. Connect the Legion to the Star Wars movie "The Empire Strikes Back" in three steps. (No trickery, metaphors, or stretches of the imagination allowed.)
I was trying to word this to get that the only answer was "Edmond Hamilton wrote the Legion, he was married to Leigh Brackett, who wrote the first draft of Empire." But several people completed a "six degrees of Kevin Bacon", including one with Kevin Bacon. Just goes to how no matter how carefully I word a question to get just one answer, you'll always find a way around it.
2. Who was the first Legionnaire to have a single-word code name (no hyphens)?
Kind of a trick question, I was wondering how many people would say "Wildfire". But in terms of publishing history, it's Superboy, in terms of chronological history including flashbacks, it's Supergirl. Had to put the no-hyphen rule so nobody could say "Mon-El" (who was actually inducted after both Superboy and Supergirl).
3. Who from the 20th/21st centuries has ever joined the Legion, become an honorary member, or become a Reservist?
L1 and L2 Superboys, L1 and L3 Supergirls, Mon-El and variations, Dev-Em, L2 Ferro, L1 Insect Queen, L1 Elastic Lad, L1 Pete Ross, Lori Morning, and the Super-Pets (not including Proty II). You forgot aboutImaginary Story Lois Lane & Bruce Wayne. I forget what the exact status of the Teen Titans is, as a couple of you mentioned them.
4. Where was the first "real" Legion logo used? (That is, one that was reused, as opposed to a one-off.)
The logo that appeared in "Adventure Comics" #300. See Todd Klein's blog which includes a 5-part Legion logo study (this appears in part 1 here).
5. Which Legionnaires or close associates have the same (or nearly so) name as a Marvel character?
According to Lorendiac's comprehensive list (v4), I see Alchemist, Andromeda, Blok, Bounty, Catspaw, Chameleon, Comet, Computo, Dyna-Sour/Dina Soar, Double-Header, Echo, Firefist, Hunter, Impulse, Inferno, Lamprey, Leviathan, Lightning Lord, Livewire, Neon, Nightwind, Omega, Omen, Persuader, Plasma, Psyche, Radion, Reflex, Shift, Slipstream, Spark, Spider-Girl, Sun Girl, Titania, Valor/Val-Or, and Wildfire.
Tom said Jewel and Gear too; Roy suggested Dreamer, Phase, and Sensor, but I'm not familiar with their Marvel counterparts.
6. According to Khundish law, a married woman becomes the wife of the man who kills her husband. Which Legionnaires were caught up in this situation, and why did the male not become the female's husband?
Devlin O'Ryan accidentally killed Firefist, husband of Veilmist. But he was too young to be her husband, so she became property of the Khundish army, until Firefist got better. Thanks to Ken Austin for the original question (as seen at the bottom of the page here).
7. Years before being flash-fried by Nemesis Kid, how did Karate Kid die the first time?
He was killed by Alex Korlo using Rakurga poison. Ultra Boy revived them using the Legion's Deus Ex Machina, the Miracle Machine. Brainiac 5, Karate kid, Princess projectra and Superboy were also "killed" at this time, as seen in Adventure #378-379. Kudos to those who remembered "Rakurga"!