If your QT doesn't have any live rock or sand, it won't have to cycle. On the other hand, your the bio material in your filter will have to get "broken in" with nitrifying bacteria before it's able to do its job removing ammonia from the system.
Anyway, I wouldn't worry at all about the nitrates, but that ammonia is definitely a problem. I would do another water change. Maybe as much as 50%, just make sure you match the temperature, salinity and PH of the new water to the old as best you can. Now, to confuse you further, there is a school of thought out there that says it's bad to do water changes to reduce ammonia, since the extra aeration from the water change will raise the PH, and ammonia is more toxic at higher PH. I'd do the water change right away, though. 0.5 ammonia is pretty much toxic, and I guarantee that wrasse is hurting.
As for the copper, I've never heard that it causes a rise in ammonia, and I can imagine why it would (not that I'm a chemist).