Just retested and we are at 5ppm. Tank has been running since March, so it's still a new tank but I believe it's cycled due to the algae growth? I will attach my attempt at better pictures but the bumps seem to have gone down in size today and most are not very visible now. There is one picture where you can see a bump sticking out. I thought it must be something contagious because we got Chelsea and did not quarantine her, noticed her eye bumps, and then she died a few days later. We also began to notice white spots on Francesca almost immediately after noticing them on Chelsea. Francescas dorsal fin is now clamped but she seems to be swimming around okay

As for tank size we are definitely open to input, but have read that 40 gallons is the minimum for two goldfish, and as these are both young we felt that using our current 36 gallon would be okay. It's not like it's a 10 gallon tank? But if a 55 gallon or something reasonable would fix this then my 13 year old would be thrilled to go tank shopping. Thanks so much for your help!
I hate to be picky but you mentioned 4 different tests ( ammonia, nitrite , nitrate and pH) so which one was 5 ppm? It matters to be explicit because 5 on a pH scale would mean there is no nitrification going to happen. 5 ppm of ammonia or nitrite is deadly to the fish. 5 ppm of nitrate is a very good reading.
Goldfish that produce a wen or fleshy " appendage" can appear to have a mold growing them until it colors up. Since you say they disappeared, they probably just colored up. It would be a lot easier if the wen was a smooth surface where a bump or pimple really stood out so people wouldn't think the fish was sick. There is another thing with male goldfish. When they become sexually mature and ready to breed, they develop pimples, usually white in color, called tubercles on their gill plates and pectoral fins. These are no usually found on the wen but it's not unheard of. No matter where they are however, there is no need to medicate.
Since you say you are new to fish keeping, I'll explain what happens in new tanks so please excuse me if I give you information you already know.
Cycling a tank means you are creating a biological filter bed consisting of at least 2 nitrifying microbes. One microbe converts ammonia into nitrites. A second microbe converts nitrites to nitrates. Depending on the pH of the water, ammonia and nitrites are toxic to the fish and if they get too high, can kill them. Nitrates are less toxic to the fish until they get much higher ( over 40ppm) where they can alter water chemistry and a fish's organs. This is a picture of the cycling process in graph form.

As you can see, ammonia goes up then down first. It starts going down when the microbe starts converting the ammonia into nitrites. The nitrites start going down when the second microbe starts converting it to nitrates. As you can see in this graph, the nitrates just go up. The only way they go down is if you have live plants that consume nitrates or you use nitrate absorbing products or you do water changes if you don't have an organic process for this. Once these two processes complete and you have had ammonia and nitrites go up then down to zero AND you have rising nitrates, your tank then is " cycled" for the amount of fish ( technically, the amount of ammonia the fish are producing) present at the time. But here's the catch, the biological filter bed is living breathing organism that grows and shrinks all based on the amount of ammonia in the water. So say you cycled a tank with those 2 goldfish, you wouldn't be able to add say, 5 more goldfish of similar size at one time without stressing the biological bed just because the tank was cycled. Overstocking too fast will cause a " mini cycle" while the microbes multiply to catch up to the new amount of ammonia. The good news is that the microbes multiply quickly but if you stress them too much, the ammonia level can become a toxic level and damage the fish. So you always want to proceed slowly. Cycling a tank with fish can take months. Usually 2 -4+ months. It's important when you start a tank with fish that you test ammonia every couple of days so that you see the ammonia level rise and then fall. With fish present, you don't want the ammonia level to go higher than .25 ppm. When it approaches that .25, you need to change enough water to get that level down but not to zero. Just to a non toxic level. When you see the ammonia start to go down without doing water changes, you test for nitrites. It's important that the combined levels of ammonia and nitrite not exceed that .25 ppm. Again, if the combined values approaches .25 ppm, water changes are necessary. Once you see the nitrite level go down, you can then test for nitrate. Some test kits will register nitrates when the nitrite level is high so no need to test nitrates until you see nitrites coming down. From that point, you want to make a regular weekly water change schedule based on the nitrate level. If nitrates approach 40 ppm in 3 or 4 days, you need to do water changes every 3- 4 days. If it takes a week, change water once a week, and so on. It's best to do water changes at least weekly not just to reduce nitrates but to reload any minerals into the water the fish may have absorbed. Goldfish are very dirty fish so don't be surprised if you are doing water changes more than once a week.
As for the algae present in the tank, Algae has no input to the cycling process. Green Algae is caused by excess nutrients combined with too much light. Brown Algae in new tanks are actually diatoms and very common in newly set up tanks. Even if you do nothing, those diatoms will die off when the nutrients they are feeding on are consumed by the nitrifying microbes.
So going forward, you need to test all 4 items and post your numbers for each. You also want to test your tap water ( or whatever water you are using in the tank) so that you/we can see the difference between your tank and your tap or what your tank is doing to your tap water.
As for tank size, since goldfish do not have a stomach per say, like other fish, they need to feed often since there is no storing food internally. Since they are primarily vegetarians, they need a lot of vegetable matter in their diet with some additional meat protein in small amounts. The problem with keeping goldfish in small tanks is that because of the feeding and their messiness, smaller tanks will go foul much faster than larger tanks. In your case, we need to figure out where in the cycling process you are because that is what it sounds like is the current problem.
Hope this helps.
