As mentioned by Sir Guppy, don't replace the sponge in the filter. Just squeeze it out in a bucket of tank water and re-use the sponge. Tip the bucket of dirty water on the lawn outside.
You don't need carbon in the filter unless there is chemicals or heavy metals in the water. If you want to keep using it you can but it only needs to be replaced once a month, or replace it with another sponge.
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The damage to the tail could be from getting it caught on something in the tank, although there isn't much there to catch it on. It could also be related to the breathing and not eating issue.
Any idea what its poop looks like?
If it is coloured and normal that is a good sign and would indicate an infection in the mouth/ throat. A broad spectrum medication that treats fungus and bacteria should help.
Is it sunken in at the belly now?
If he has sunken in at the belly that is an internal protozoan infection and Metronidazole might help.
Has it gotten fat at all since this started?
If he got fat rapidly when this started, that is usually an internal bacterial infection and there's not normally a cure.