I would submit that it takes three to eight weeks for a filter to build up the necessary bacteria (good kind) to consider the tank cycled, and agree that if the tank has been running with fish for less than this time,that replacing the filter cartridge could have destroyed a good portion of the biological filter.
Gravel does indeed also hold beneficial bacteria ,and getting too aggressive in cleaning it and replacing filter cartridge at same time could also remove beneficial bacteria.
Best to clean filter as mentioned in dechlorinated water or some of the old water you remove during water changes and stick the cartridge back in the filter compartment. Should only need to replace the cartridge when it begins to fall apart from the numerous rinsing's.
Feeding too much can also cause ammonia levels to rise in aquariums as well. In any event,, water changes using a good dechlorinator such as PRIME will detoxify ammonia that may also be present in your tapwater if the city where you live uses chloramines to treat the water.
Some water conditioners are considered full function and PRIME is one of them.
Seem to recall you mentioning that filter cartridge for your filter uses carbon and zeolite. Products like zeolite and ammolock can affect test readings for ammonia by rendering false positive or so it has been reported. Would also submit that if using zeolite or ammolock that I would NOT use salt in the aquarium. Some products such as these ,absorb ammonia and can be recharged and used again by soaking in salt solution which causes the material to release what it has absorbed. If using salt in an aquarium with these products,it could cause ammonia that has been absorbed to be released back into the tank if salt is also used in the tank.
I am doubtful that you destroyed all of the beneficial bacteria but would keep up with testing and water changes until ammonia levels are once again at or very near zero. I would also reduce feedings a bit and see that no one else is also feeding the fish. If the tank was cycled at some point,,it should rebound without too much trouble but do keep an eye on parameters.
In the future,, try to space out cleaning of tank. perhaps rise the filter material out one week,the next week clean gravel with siphon,and perhaps clean glass with clean paper towel folded into fourths. try not to clean everything at one time.