It's all related to SeeD rank. In the Ifrit timed challenge, the less time you have remaining when you beat him, the better your SeeD rank becomes. For the Dollet exam when you're running back to the shore, the faster the better your rank. For the Timber Owls train mission, time doesn't play into it but if you do it without failing and choosing to try again, you'll get plus one to your SeeD rank.

For the missile base, I've found if you select 20 mins, your SeeD rank goes up one once you beat the Iron Clad. I've also discovered that not using GF (throughout the game) gets your SeeD rank nice and high and less prone to dropping. I don't know what happens if you pick 10 mins though.