Is DJ appropriate?
As you have discovered, those fish were a heavy load for an uncycled tank. (Please look here and elsewhere for The Nitrogen Cycle.) Depending upon the authority, it takes 30-60 days for a tank to cycle. In a sense, if one is gradually adding fish, say a couple guppies the first month, the five tetras the second month, some more tetras a couple of week later, the ability of the aquarium to break down the very toxic ammonia to less toxic nitrite to yet less toxic nitrate (boy I keep confusing those last two) will gradually increase.
Even though it slows down the cycling process, it is a good idea to get used to doing 20-40% weekly partial water changes. That removes some of those waste materials gathering in the tank. Your cloudy water was a bacterial bloom caused in part by the inability of the aquarium to process the fish waste - especially of the goldfish. That is a normal part of the process, but one should be changing water and really careful not to overfeed. ("More aquarium fish have been killed by overfeeding, yada-yada.")
They don't too often tell you this in the shops, but you need to keep a supply of water for a week or so before changing it in. If your water department is adding chlorine AND ammonia (= chloramine), you don't just want to add a dechlorinator. That will release the very toxic ammonia into the water and you may get a sudden die off of fish. If the stuff sits for a few days, the chlorine will naturally, though slowly release into the atmosphere. (Actually there will still be some there after a week, but it will be less.) If it is bonded to the ammonia, it will not release. One of the nicer arrangements is to SLOWLY run it through one of those carbon block filters or faucet filters designed to take all sorts of nasty stuff out of the water.
What are you trying to neutralize in the water? (Call up and ask your municipal water department if need be.) What water conditioner are you using? There are a lot of good ones. They do need to fit the need though.
And there is one conditioner, which guarantees that a person will never have to change the water in their aquarium again. Yeah - and the aquarist will never have to take a bath again either! ;)
Don't do a tear down during that cycling period, unless the tank smells like something that died. What you do is force the nitrogen cycle to start all over.
If there is a lot of either chlorine or ammonia in the water, one of the things they will do is burn the fish's gills. Their immune system can be crippled and they mysteriously die sometime later.
If you are going to mail order fish, this would be better than when the weather gets colder. Although terrifically expensive, go with the one-day delivery in winter.
Problem: as your now know from looking at the Quick Links on ich, it take a couple of weeks to treat and banish ich from a tank. What safe tank would you put you mail order fish (a considerable investment probably?
I know that has got too sound really crabby. The shops and mail order people will not mind selling you new fish every week. You may want to consider what is best for the fish though.
Just out of curiosity, was the fire tail tetra nibbling on a dying fish or was it actually killing a healthy fish? What scientific name goes with fire tail? (I lent my Baensch Atlas to a neighbor kid.)
25 is a great age to be. Several of the GL members are in your age group. There are quite a few younger than you. Also there is a good sized group of adults a little older than you. Oh yeah, and one geezer and a couple of geezlings.
All the best!
uncle geez