In short, I don't get how the medal-awarding system works. I understand the story-related medals like the Epilson Pegasi liberation award, but I don't understand how the system for rankings and aces is set up.
1. I've finished the whole Freespace 2 campaign and I'm still only a Lt. Junior Grade. Why?
2. Does the score you earn from one campaign transfer into other campaigns? Does your kill count carry over?
3. How many kills do you need to get a triple ace? (I've got 350-ish after the last few campaign levels, but since there was no debriefing I didn't get a medal).
4. What determines my rank?
5. Is there any benefit to being a higher-ranking officer besides owning more bling?

Your rank is determined by your point score, which in turn is determined by your skills, and a multiplier based on the difficulty. Your score and kill count both carry over, but it is highly recommended that you create a different pilot for each campaign, so that if something goes wrong, you won't lose all your progress.
To get the Triple Ace, you need >350 Kills.
Your rank is useful in Multiplayer; if you create a game online, you can exclude people based on their rank.

Okay then. What effect, if any, does the use of cheat codes have on ranks?

in short nothing...
If you activate cheats for a mission, the score doesn't count

I did some testing... if I activate cheats on a mission, my score still goes up depending on the number of kills I made.
Also, I've learned that you need 500 kills to make Triple Ace.