It is quite possible that the tank was suffering from old tank syndrome. This happens when a tank is neglected and few water changes are done. The fish use up minerals in the water and they secrete organic chemicals, which build up. They also excrete ammonia, which is turned into nitrate and that also builds up due to lack of water changes. The nitrate and some of the organics are acidic so any carbonate in the water is used up and the pH falls. Tank water will evaporate and if the previous owner was in the habit of topping the tank up rather than do water changes, this increases the mineral content, especially calcium, which makes the tank water harder and harder.
The result of all this is that the nature of the water in the tank becomes very different from the water that comes out of the tap and a big water change can harm the fish. The tank water was also likely to be quite different even after the big water change compared to the shop tank. And using RO water would have changed the water even more as that has no mineral content, I'm afraid.
Neon tetras are not very salt tolerant and as you put that in before you got the new fish, they would not have been very happy when they got into your tank, I'm afraid. The API salt is just an expensive version of the salt you put on your food, plain sodium chloride.
You can use RO water for freshwater fish, but like with marine fish you need to add the right kind of minerals back in. If you want to use RO for this tank, you will need a GH and KH tester in addition to ammonia, nitrite, nitrate and pH. The first think is to find out the prefered GH and pH of the fish you want to keep then add remineralisation salts to each bucket of new water at the dose rate to get the GH and pH you want. Without the testers you will be flying blind. Remineralisation salts -
Tropic marin or Kent RO right (liquid or solid)
Anything with claws will try to catch fish, and the bigger the clawed creature is the more successful it will be. The lobster (possibly a crayfish of some sort) does need to go. The shark needs a bigger tank, so either rehome that or buy a new tank.