How big is the tank? Tank size needed for these two species' is vastly different so the size of the tank is likely to be the deciding factor. A single male Betta will be happy in a tank from about 30 litres upwards, but not too big. A fancy goldfish grows to a very large size around 8 inches long and very bulky. Goldfish need to be kept in pairs or groups as they are a social species, so for two, the minimum tank size is 180 litres.
The Siamese algae eaters were extremely effective for me, but they do need to be in a group of at least 4 so if you like the look of them I would do that. If the idea of having 4 quite large fish in the longer term doesn't appeal, go for the liquid carbon treatment. It will take longer as you'll have to spot treat a bit at a time to avoid overdosing the tank. I wouldn't combine the two methods.
Hmmm.....something's off with the test results. In a cycled tank you should never see 0 nitrates as nitrate is the final product of the cycle and only removed by water changes. Maybe your test kit is out of date? What test kit do you use?
There are a couple of potential solutions that will work in conjunction with getting back on top of maintenance and regular partial water changes. The first, which I used myself is a group of Siamese algae eaters. They cleared masses of it from my 6ft tank in just a few weeks, but only get them if you really want a group of them long term and make sure you do get true Siamese algae eaters as some shops will mislabel the Chinese algae eater or the Flying fox and you don't want those. The other solution that works well is to spot treat the algae with "liquid carbon" such as Seachem Flourish Excel. Don't overdose the tank though. Just drip it onto the algae (preferably when it's exposed during a water change, and it will gradually die off - it usually turns pink as it dies.
That actually looks more like black brush algae. The most common cause is overstocking and/or inadequate water changes, often combined with too much light. Can you post tank size and a list of stock plus test results for ammonia, nitrite and nitrate, please and how long your lights are on each day?
It's actually very unlikely that a lack of an air pump caused the deaths. Many fishkeepers don't use air pumps at all as the rippling from the filter return is plenty to give good oxygen exchange. As you say you added a lot of fish to the new tank, that is most likely to be the reason for the deaths. Adding too many fish at once causes ammonia and nitrite to spike and poisons fish. Also, did you move over the filter or filter media from the old tank? If you didn't then you had no beneficial bacteria to process fish waste and again, ammonia and nitrite poisoning would be the result.
Now, when it comes to adding new fish the first thing we need to know is whether your water is hard or soft and what the pH is. This is because not all fish like the same type of water - guppies and platys like hard water, tetras like soft water. You can find your water hardness on your supplier's web site if you don't have a gH/kH test kit. For the pH, test a sample of tap water that's been left to stand for 24 hours.
What kind of fish do you have? I'm going to guess they are livebearers of some kind (Platy, Molly, Guppy, Swordtail). If that's the case it's quite simple: the females can store sperm for up to 6 months and then choose when to become pregnant. This means you can have babies appearing in a tank that has only females or even only one fish. The babies were most likely hiding in the plants for some time which is why you didn't see them. The additional babies will be offspring of the first ones you found as many livebearers are mature enough to breed at 3 months old.
Cleaning everything probably removed the beneficial bacteria that actually filter the water and convert ammonia via nitrite to nitrate. However, the underlying cause is far too many fish for the size of the pond. 220 litres is only big enough for a couple of goldfish. This article explains how much space goldfish need: https://www.fishkeeping.co.uk/articles ... -size-life-expectancy.htm