Yes it will nuke the live rock. Copper is deadly to almost all inverts, vertebrates have developed bodily functions to handle it to different levels, though it is still ultimately toxic to fish. For that reason I don't advise prophelactically treating fish with copper 'just in case' - use it, in QT, when you need it.
QT is always a good idea, no matter how good the lfs. If they have a lot of fish in and out, and/or use a centralised system, you have to assume there is ich, or something else in there. It's just a fact of life. The only exception I have is that for example, I have an lfs I wouldn't quarantine from. He doesn't get many fish coming through, he doesn't have centralised system, and his fish often sit, in the same tank, for a month plus. Effectively he quarantines for you, but obviously this is the exception and not the rule. (he doesn't do marines for the money incidentally, just to have a 'complete' range).
I doubt melafix is going to do it for you, and I don't trust most natural cures - is melafix, a general cure, specifically going to hit ich, brooklynella or whatever but not hit your other inverts? Get real! Most natural cures work by promoting the slime coat so the parasite can't stick, or garlic makes it taste bad to the parasite(?).
Here's my QT regime, which I've also used with obvious changes in freshwater. Setup QT tank with maintank water + some fresh, plus live rock and cheap skimmer if you have one. Also a sponge filter if you have one. Get fish,float bag to start to match tank temp, add little bits of tank water. When you're ready to tip fish in DON*T - if possible, freshwater dip whin pH corrected water to avoid possibly fatal pH shock. Net fish out, dispose of water from lfs. Watch fish in dip for up to 10 mins, but many won't go this long. You've now gotten rid of any nasties in the water, and the freshwater dip will deal with many more. Leave fish alone in QT, feed as best you can (this is why pieve live rock , hopefully with algae helps). If fish gets sick, remove live rock to a smaller bowl, or 2 litre yoghurt pot NOT main tank, because it too needs qt. Now treat QT tank. Filtration - assume it's gone. Maintain water quality with water changes and skimmer if you have one - cycling is irrelevant here. If you use copper, use a test kit, and beware , fish are variably sensitive to copper, and you don't want to kill them. This works for me, and once you've killed a few hundred pounds worth of fish in one go, you'll do this too.