I think the "green disk" that is being referred to would be a fish louse or Argulus. They look like little green flying saucers that stick on your fish and suck their blood.
Actually, there are only a very few parasites that will survive in a fishless tank. Most parasites require a host as do most fungus and bacteria that can hurt the fish. Without a fish in the tank, they will starve out.
I personally think, as "BigRed" has said, that most likely the fish jumped because of poor water - perhaps combined with parasites. The others may have had other problems and, in a cycling tank, simply were not strong enough to survive the toxic cycling tank.
I personally am not a fan of cycling a tank with fish in it. I guess I am just too lazy. I do not like the daily water changes and the worry and the stress, added to the fact that I strongly believe that exposing fish to high levels of ammonia and nitrite can permanantly harm them - leaving them open to a whole host of difficulties down the road.
This said, there is a way you can cycle your tank, and starve out any bacteria and parasites that might be in there. This would be a fishless cycle.....
http://www.kokosgoldfish.com/FishlessCycle.htmlYou do ont have to start again or anything. The 2 + weeks it will take to get a robust cycle going will be sufficiant to starve out any nasties that may have been in the tank.
This is the best of all worlds for you do not have to work to change the water, no fish are harmed in the process and you will have the 2 weeks of an empty tank that can starve out flukes, lice, ich, etc. It is all a good thing.
Then when you do bring new fish in, give them a good salt dip, place them in the fully cycled tank and away you go!
I am so sorry your fish died. It happens - but is always very sad - particualrly for a child who does not understand why things happen. But you should be able to set up your tank, and enlist the help of your three year old in doing it - so you will have a happy bright environment of healthy fish to watch.