Failure

So if you fail and try again, are you still legible for the rewards?

So long as you don't finish the 'mission' as it's a shadow mission, yes you can quit the mission and talk to the npc again.

Once you 'finish' it by talking to the correct NPC, the event gives you that reward, based on the time and doesn't give it again if you get a better time.

Does the time tick down after starting the mission or after closing the intro dialogue?

I expect the time to start the moment the mission starts. I don't expect Devcats to code it for the user's sake. Especially when they are determined to give the 2nd hint at 3 minutes in and 3rd at 5th minutes in.