Don't change the media while your bb is trying to become established, and you never want to change everything at once or you'll crash your cycle. The sponge can be replaced every 6 months to a year. The carbon will need to be replaced at least monthly if you choose to keep using it. Some people replace that biomax insert never and others will replace around every 6 (not the same month as the sponge is replaced). Once the tank is fully established, the sponge and biomax can be rinsed in old tank water along with the weekly water change to keep debris from getting built up.
You need to invest in a good drops test kit for ammonia, nitrite and nitrate and be doing daily tests and water changes according to what comes up. First you will see ammonia spike. Eventually ammonia will go to zero and nitrites will spike. When nitrites become zero and you will see some nitrates you have completed your cycle. Nirtites in small amounts are okay but should be kept below 20ppm. Weekly 20-50% water changes should be enough after the cycle is completed but you really need to monitor carefully and make far more frequent changes until this happens. Here is a guide: http://www.bettafish.com/showthread.php?t=47838
Here's a test kit most people use: http://www.petco.com/product/103685/...t=OnSiteSearch
Fin rot during fish in cycling is pretty common because they will be exposed to less than ideal conditions including ammonia. Once you get your cycle completed it will be easier to keep his water clean.
As long as the fins are no long receding he should be okay. If they're continuing to recede you may need to stop for treatment. If you don't already have an established quarantine tank you can use, you can get an acrylic 1-2 gallon bowl or kritter keeper of some kind, fill it half full with water and float him in that in the larger tank to keep it warm. In this container you can treat with 1 ml per 1 gallon stress coat to help fin regrowth and also 1 tsp per gallon fresh water aquarium salt, predissovled. Any time you make a water change you will need to add as much stress coat and salt as water you take out.. so if you have 1 gallon of water, do a 50% change you put 50% (or half tsp and half mil) of the salt and stress coat back. You can do this for up to 10 days. Meanwhile you will need to find another source of ammonia to feed the cycle. Fish food works, as does a drop or two a day or pure ammonia from ace hardware, but be aware you will need to test and likely do a larger water change in the tank to make it okay again before adding him back or his fins will only start to rot away again.