As of now, he doesn't swim at all ): Since I don't have a filter, or anything to help aerate the water, I was told that stirring up the water and putting bubbles in will help. Tweek, my afflicted goldfish, used to swim around with the current, but now he just floats. One of his fins works, but it doesn't help him at all. ):

Hi.  Your fish is beautiful.  Sorry he is sick.

Your tank is cycling, which takes time and can be very hard on the fish. You seriously need to get a filter soon. An air stone would be nice too. Goldfish are always creating ammonia and will be poisoned if the tank hasn't cycled. You can get some Prime and add that for now, until your tank stabilizes.

He could have a swim bladder issue, be constipated, or have a virus or bacterial disease.  Most problems are caused by poor water quality or a bad diet.

What should I get for fin rot? Because I believe both fish have it. The gold one, its on his tail, and my other fish has a small spot on her tail, but she's still fine.

The bacteria does not live in the water.  You can change as much water as needed but there is never going to be a cycle without a filter on the tank.   Goldfish need great filtration because they produce so much waste.  Gravel is not necessary and I only have a very small amount in my own tank, doesn't even cover the bottom now.  Gravel is where a lot of nasties hide and can cause problems honestly.  Once you get a filter you can begin your cycle.  If your filtration system only has one pad then you can usually fit a second pad in the back part of the filter.  The reason I would do this is because you never want to throw away all the good bacteria once its on the filter sponge/pad.  I have 2 pads plus 2 bio-wheels and a sponge filter on my tank but its a much larger tank so I don't have to worry about this part but you may with a smaller system.  If you have the 2 pads then you can replace one at a time once they are really bad, mainly just rinse them in TANK water only every week when doing weekly water changes once your cycle is done.   During the cycle you are going to need to do huge water changes and use PRIME as your water conditioner to neutralize the ammonia and nitrite.  I would do around 70% water change every other day at least till your tank has cycled because of it being too small and having 2 goldfish in there.   Melafix will help heal the fin rot but I'm not sure if they can handle any medication at this time.  First get the water as clean as possible and get the filter then you could medicate but don't over it.  Something harsh can do more harm than good.   You have no cycle going right now so don't worry about that part. 
Good luck.