First things first, change the water, change a lot of the water, then read on.

Do you have a liquid drop test kit for the water? You'll need to test at least the ammonia and nitrites daily until the tank cycles, and yes a 5 gallon can cycle with two tiny goldfish, but 20+ gallons will be better in the future. Until the tank cycles you should change 50% of the water daily seeing as its such a small tank. Don't clean off the tank walls yet though.

Changing the filter materials is unnecessary until they are disintigrating. The beneficial bacteria you need will live on this material. You can swish out the gunk in a bucket of old tank water every now and again after you clean the tank, but not before the tank is cycled. I squeeze out my filter sponges if they get clogged and then maybe twice a year if they don't. Carbon is completely unnecessary. It only is active for a few days. However, since its so porous it makes a good place for the bacteria to live, so you can just keep it in the filter without changing it.

Gravel is fine. Its another place for the beneficial bacteria to live. For such a small tank you can spend a dollar and get a turkey baster to suck out the poop and dirt from the gravel during cleanings.

After your tank is cycled you should not treat bacterial infections in the tank. A large bucket works for a treatment tank for a goldfish because you can change it 100% daily. Any antibiotic treatment will kill off all of the beneficial bacteria in the tank, and possible kill the fish.

Fungus on the tail is best treated with cycled water. You can add all the medicine you want to a tank, but if the fish has a poor immune system from stress and poor living conditions and so on, its a lost cause. Considering there's just white on his tail, I personally wouldn't give it any medicine at all. Most medicines the store gives you will do much more harm to the fish than good. Just work on good water conditions. These look like orandas so they will sometimes get a little white on the head as their 'wen' (the head growth these goldfish have) grows, but it won't be on the tail so it will definately be a fungus right now.