Ok, I need to make sure I'm clear on all of this before I suggest the meds. The paradise fish... the lump on the mouth... does it look "fuzzy" or does it look more like a chunk of cottage cheese stuck to his lip or growing out from it? The sick cory.. what exactly is he displaying for symptoms other than inactivity?
If your test results are accurate I see no problem with the water quality in the paradise fish and cory cat tank. Please stop the Melafix treatment today, add the carbon to your filter to remove remaining traces of it, and perform a 25% - 30% water change each day for the next 2 days. Melafix is not going to fix this problem but it is not safe to mix it with the meds we will be discussing/using. We will need to wait at least 48 hrs after you begin the process of removing the Melafix before we can add the proper medication.
Now, lets address your neons. Neons should not be in any size tank with a rainbow shark. Those fish simply are not compatible. It was very likely stress that killed the neons, and as time moves forward the rest of them will die also if they remain in a tank with a rainbow shark.. either from stress or from being eaten. As the shark grows so will its aggression level and territorial issues.
I am planning to use a medication in the paradise tank that will treat all of those fish together safely. You may need to remove any live plants during the course of treatment, but I see no need to use the quarantine tank for them. My suggestion is to take that 10 gallon, set it up, borrow some filter media from the current rainbow shark/neon tank and get it filled up and running... add heater and watch for temp to stabalize as same as what's in the shark/neon tank now. Basically the idea is to match the conditions in the shark/neon tank as closely as possible, then move the neons to that other tank. This will buy you some time to figure out who is going to go where, stay where, etc. without the issues of rainbow shark/neon incompatiblity killing neons steadily until they are gone. I am wondering, how big is the tank they are in now?
I will be around for another few hours and then probably again late tonight. We have time so I am in no rush... isn't much else I can do for you for the next 48 hrs or so anyways. I will get you the name of the proper meds once I am sure about what you need to be using... after I understand for sure what you are describing.
Hang in there... once we get it all straightend out things will get better, easier, and you can get back to just enjoying your fish.