Really hard without a picture of the tank. Normally fish will die in a tank at about three months due to ammonia poisoning. The fish and corals you added too soon for the bacterial population to catch up. This is very common at the 3 month mark.
Ammonia won't show up on a test kit after the fish die because the bacteria will process it pretty fast, just not fast enough to keep the fish alive.
A tank is not instantly cycled as soon as your test kits read there is no more nitrite. That just indicates that you have enough bacteria to take care of a dead shrimp or whatever you used to cycle the tank with.
It takes about 3 months for the bacteria to grow enough to add more than about one small fish.
What did the fish look like as soon as you found them? Did they have their mouths closed or wide open?
They "probably" did not die from parasites as you would have noticed that and fish "never" die from lack of oxygen unless the water was visibly cloudy.
I don't think going fallow is the answer. You need life in there to keep the bacteria multiplying and an empty tank will cause the bacteria you have to die back. The corals don't contribute enough ammonia to keep the cycle going.
I would add one small fish and nothing else for a few weeks. Then if all is well, add another small fish.
Good Luck