I think for trials, you get the echo if the party manages to stay alive 3 minutes after engaging the fight. 5 minutes seems to long in my experience.
Each echo step is 10% for HM trials, Chimera and Hydra, 5% for EX trials. There is no echo for MSQ trials, nor for Hildibrand's quests trials.

Regarding Coil, it's a flat 20% for the Binding Coil turns 1,2,4,5 (no echo in turn 3) and Second Coil turns 1,2,3,4. It's only a flat 10% in Final Coil turns 1,2,3,4. That being said, we can imagine that Final Coil echo may increase to 15% and then 20% in the future (it has been that way for BCoB and SCoB).