Hi IrieJar,
Do NOT replace your filter sponge. Do NOT wash it in tap water. All that hard work cycling your "tank" was really about cycling your "filter." The two bacterial populations you should have growing in there now are attached to all the surfaces in there and the most important one to you is all that sponge surface. That mess in there is your friend.. well, eventually it clogs up the filter and slows down the water flow, so at that point you have to do something, right? What you do is wait until a water change. As your siphon is cleaning the gravel and water is siphoning out of your tank, you direct it into a bucket and then lift the sponge out of your filter and lightly hand-squeeze or swish it around in the tank water to release most of the debris. Its very important that it is tank water because there is not chlorine to kill the bacteria! Don't over-do this rinsing - you want to preserve most of the bacteria. When doing this operation on your filter, don't get side-tracked - you don't want any media that has bacteria to get dried out and you want the flow of tank water through the filter to resume in a reasonable time so that the bacteria can get their ammonia food and oxygen - they like the flow.
Someday, probably a long time off, maybe years, the sponge may begin to break down and you'll decide to start replacing it. You do this by getting new sponges and then taking scissors and cutting both your old sponge and a new sponge such that maybe one-third new is added at one time. That way 2/3 of the old sponge with bacterial content is retained and keeps running as your bio-filter. You can even take the old third and float it in a cloth bag in your tank for a while to keep the full bacterial load working while the new sponge gets cycled, but this is probably extreme.
About the carbon: you don't need it for normal operation. Do some searches on TFF and you'll quickly learn the thoughts about it. you can replace the carbon area of your filter with some more sponge that fits the space. That will increase your bio-filter capacity - always a good thing (don't forget, wild fish often have -acres- of biofilter curtesy of mother nature.)
Hope this helps explain it a little, ~~waterdrop~~