As I see it we are at a stalemate. But, I feel sorry for the fish...so I would like to offer an olive branch the original poster..
A ten gallon is just too small, and aquarium salt does not a brackish tank make..
But we can help you keep them alive until you can upgrade to a better situation or re-home them.
No need for bio class- The Nitrogen Cycle. Fish produce ammonia. With out a cycled tank the ammonia is not converted into less toxic nitrates and nitrites...See, like a circle..ammonia to nitrites to nitrates..
Ideally you would have a liquid test kit to help you monitor the levels..You want 0 ammonia, 0 nitrites and <10 nitrates...
What "does" the cycle: Bacteria! Bacteria live in your filter media..so you should not change it. Yes, the filter makers want you to buy new ones every week..but really they should simply be cleaned lightly and changed rarely.
Until you have a cycled aquarium, you should change the water a few times a week. Asked your friend for used (wet) filter media to start the cycle in your tank. Then buy a proper test kit so you can monitor your levels.
Remember that fish keeping is a science, not a hobby like knitting. You have to learn to do it right. I know you mentioned a 10-13 year old gold fish, but gold fish can live substantially longer than that.
Everyone must start from the beginning. Some people start out thinking they know everything and end up killing fish, please do not be one of those people. Remember this is a science. It takes skill to do it right. You must learn those skills.
I know you think your friend knows what she is doing, but plain and simple she/he does not. Fish keeping is full of people who "think" they know what they are doing...but no one who knows what they are doing would advise you to change your filters every other week Because doing so takes away your bacteria and will kill fish...
Now, what else do you not understand or need help with. Ask and we shall help.