Hi there, what a shame to have to start this way, but rest assured you are not the first and will not be the last to get this sort of 'advice and help' from the fish shop
The good thing being you're in the right place, listening to advice and obviously willing to learn and do what you can to have a sucessful fishkeeping experience, and we will give you all the help that we can to get you through this.
However there is no point sugaring the pill as it were, I always take the approach of honesty, apologies if this comes across as blunt but I would rather you go into this with your eyes wide open than harbouring false illusions.
The neon does not sound good, all his tail eaten away is a very bad thing, you may find you need to choose to euthanaise him, don't rush in we'll try to treat first but prepare yourself for this possibility, we can also help you on humane euthanasia if it comes to it. Can you try and get a close up picture of this so we can diagnose the problem (it will certainly be linked to the cycling tank however this could have triggered a couple of different problems so it's important to identify exactly what disease/situation he is in)?
As WD has said, the serpae's hiding away there is not a good sign either.
Your immediate focus should be water quality, keep doing water changes and monitoring the water, you should try to keep ammonia and nitrite as close to 0 as possible, definatley below 0.25ppm. Just do as many or as large water changes as you need to to achieve this.
The ammonia in your tap water will be a limiting factor, quite a few people have some and it's not a long term issue. Once the tank is mature it will be able to handle the small amount of ammonia added in when you do a water change, the issue is when the tank is not cycled the filter can't handle the extra ammonia. So it's a case of monitoring tap and tank water's ammonia, obviously if you have 0.5ppm of ammonia in the tap water then no amount of water changes will get it below 0.5 in the tank until it's cycled. If you get to the point where the tap water has a higher ammonia content than the tank then water changes may become redundant, HOWEVER you must also factor in the nitrite, when you get to this point the ammonia eating bacteria will have grown to a point that they can handle the extra ammonia added in from the tap water, or at least some of it. If you have 1ppm of nitrite in the tank and 0 in the tap then the purpose of the water changes switches from being to reduce ammonia to being to reduce nitrite. So I think that makes sense.... let me know if not!?
Keep a record of all your test results and what you have done each day, fish added, water changed etc etc. If we need to work out where you are or diagnose any problems the more info you can give us the better so this will be a considerable help. Lots of people keep a small notebook or a spreadsheet with test results.
As WD said, read some of the links in my sig, most important one for now is 'whats cycling', I'd also recommend you read through the step by step guide to setting up an aquarium, I know yours is already set up but it covers a lot of points so there may be some things you have missed.
Don't try tinkering with the pH just yet, but do monitor it, the good thing is your tap water pH is 7.4 which is absolutley fine, as you're doing lots of water changes it should keep coming up to near that level. If it drops close to or below 6 then post up and let us know, we'll advise from there.