Sorry, metazoan is just a technical term for multicellular life, so worms, arthropods, etc.
For TTM, a number of people's fish are damaged by the process - either the frequent netting and moving, or people not realizing how fast ammonia can build up in the containers. In terms of stability and freedom from stress it would go: DT > QT >TTM containers.
Timing with TTM is so critical, and people sometimes make mistakes. I also have a strong suspicion that Amyloodinium (velvet) tomonts may ride through on the fish's gills, and never actually drop off of the fish, so TTM doesn't screen them. I have no proof of this, but I think it is likely.
Problems with copper revolve mostly around people starting too late, running lower than normal doses, taking way to long to achieve a full dose and improper testing (API test kit is difficult to use for example). I haven't seen true copper toxicity myself in literally thousands of fishes I've treated in the two decades since I stopped using copper/citric acid mixes. I have seen some issues with organically bound coppers not acting fast enough to treat acute protozoan infections - lets say the fish are within 24 hours of death, but the copper is going to take 72 hours to effect a cure....that isn't going to end well. Not sure TTM would save them either though.
Jay