Looking at the manufacturer's website, it seems as though the filter cartridge contains a sponge and another section with black and white bits. (it has been a while since I last looked so I wanted to make sure it was still the same

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The black bits are carbon and the whitish bits are zeolite. Carbon adsorbs things onto its surface and gets full in a few weeks. Zeolite absorbs things as well, but unlike carbon it also absorbs ammonia - and like carbon it gets full and then stops absorbing ammonia. This is why they tell you to change it regularly.
You don't actually need carbon full time as its main uses are to remove medication after treatment has finished and to remove the brown colour that comes out of bogwood. Zeolite also absorbs medication (which is good once treatment has finished) but you don't want to continually add new zeolite as it will grab the ammonia before the bacteria have chance and they will starve. You can do one of two things.
You leave the carbon/zeolite mix in there permanently. They will both become exhausted (they probably are by now) but if you don't change them you won't be adding fresh zeolite. Should your fish ever get sick and you need to add a medication, they should be well and truly saturated and not remove any med.
Or you could remove the part of the cartridge that contains the black and gravelly bits (force it open if necessary) and get some filter sponge, any make, and cut it to fit. That way you won't have any carbon or zeolite, but twice as much sponge.
I haven't used carbon for years, not since I found out I don't need it.
Sponges need replacing when they fall apart or won't go back to shape when you squeeze them. If you decide to replace the carbon/ zeolite with another sponge they can be safely replaced one at a time with at least a month between. The rocks on the bottom of the tank which are the main home for the bacteria should be replaced a few at a time when they start to crumble, which should only be in a few years' time.
Now that you've got the ammonia and nitrite nice and low, be guided by their readings for water changes. You can probably do smaller daily changes to keep them both below 0.25. Once they go up slower and slower you can space the water changes out a bit. When you reach a day where you haven't needed to do a water change for a week and the readings are zero every day, that's when the tank will be cycled and you can do weekly maintenance water changes of 25 to 30%.
Your tapwater nitrate sounds nice and low. This is the least accurate of the testers - you need expensive lab equipment to get really accurate results - but it is good enough to show when your nitrate level goes up. As the ammonia and nitrite stay at zero between water changes you will find that nitrate goes up. This is the reason for doing weekly water changes once the tank is cycled, to remove the nitrate and other things like fish hormones that also build up.
For now, yes you can feed the fish once very 3 days but as the ammonia and nitrite readings start to drop (or at least slow down) increase to every 2 days, then every day.
You will get there
