Yeah, I'd say to take a step back and think about starting over/relearning the basics you need for keeping a saltwater tank.
With regards to the QT:
-Yes, the quarantine tank needs cycled.
-Nitrite is essentially harmless in saltwater tanks.
-As long as the tank is quickly processing/can quickly process meaningful amounts of ammonia, it's cycled.
-A lot of people cycle their QT's using filter media from their tank, as the media is already cycled and can bring the needed bacteria with it - others may prefer the sterile method using bottle bacteria. As mentioned, the water from your display tank will not contain enough nitrifying bacteria to make a meaningful difference any time soon with cycling; it would probably take a more than month or two with just the water to get the tank properly cycled.
-If you want to use bottle bacteria, Fritz TurboStart 900 is the recommended bacteria, as it has the most evidence of effectiveness - BioSpira and Dr. Tim's One and Only for Saltwater are the next suggestions. Those three brands work, others seem not to, so I'd stick with one of those three (Dr. Reef, Taricha, Dan_P, and others have done some great work on testing these). It's recommended that you add ~3x the suggested dose (i.e. add 3x more bacteria than the bottle tells you to add), and then your QT is ready ("instant-cycled").
-Regardless of cycling method, for the common diseases in the hobby (other than uronema), proper, medicated QT will eliminate the disease. So cycled filter media is fine.
-As mentioned, don't use calcium-carbonate based stuff (rocks, sand, etc.) in the QT, as it'll mess with the copper levels; PVC pipes are generally used in place of rocks to prevent messing with the copper levels while providing hiding spaces for the fish. If sand is needed/wanted (like with sand sleeping wrasses), people have used a bowl of Quartz/Silica sand in the QT without issues.
-It may depend on the medication, but, yes, after a treatment, you do want to change the water to remove the meds from the water in the tank (this is discussed in the QT protocol below for the copper at least - I'm not sure how necessary it is for other meds, but I'd personally probably change the water after the full treatment is completed regardless to be safe).
-For current, proper QT protocols, see the thread linked below:
2024 Quarantine Procedures Jay Hemdal David Scarborough Protozoans (Cryptocaryon/ich, Amyloodinium/velvet) and Metazoan trematodes/flukes are by far the most common parasites found on newly acquired fish. A carefully managed, proactive quarantine process can effectively eliminate these...
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