The carbon really needs to be changed weekly, after a week it isn't doing its job anymore. The slime is growing because of nutirents that your PWCs will take care of. If you are doing a 10% PWC
bump it up to 15% for a while. Remember to mix your SW
up at least 24 hours before use.
I don't have a UV
sterilizer on my tank, though I have thought about adding one. There are pros and cons to them. I just can't find enough pros for my personal tank to add one at this point.