That that has happened when the ammonia is near zero and nitrate is rising is that you have some amount of bacteria that can process some fish waste. This does NOT mean that the tank is cycled, as in done. It is in the process of cycling. You can progress with the tank as long as you are smart and go slow.
I don't know if the fish died because of elevated ammonia, but you can likely bet that those 15 fish did produce more waste than your tank could currently handle. The tank should quickly catch up, but death could have happened. Also, and this is very important that many so-called cycle experts do not point out is that fish DO NOT have to die to have gill and other damage from high ammonia - think of respiratory damage to humans that just shorten life or give cancer, but not death. Fish don't have 1-800 lawyers on during mid-day Judge and talk shows to represent them in class-action lawsuits when hobbyists stock too fast.
Even though we do not know, fish gasping for air is really bad and a good sign that gills were damaged, probably from ammonia. Not all fish react to ammonia at the same levels, so it is normal for some to die while some have gill damage and others are not affected much at all. If you lose some more fish in the next few months for no reason, this ammonia exposure could be the reason, so don't try and chase any other explanations if you cannot come up with any simple ones.
There are sensitive fish that can be harmed by high nitrite, but we are talking levels likely that you never hit. Even then, nitrite is more of a slow death thing and not instant.