That little system is really only about big enough to be a learning system. Please be sure to watch the temperature on such a small system as it will fluctuate quite a bit. Luckily goldfish are tough. Be sure to talk with the guy in the aquarium shop about treating source water but if you intend to eat any veggies from the system, you want to be careful of any water conditioners sold for aquarium since they might have things in them you don't want in you food. The guy in the aquarium shop will probably know what the local water supply uses to treat the water. If it is just chlorine, you can simply set up the system and let it run for several days to outgas the chlorine. If your water is treated with chloramine, you probably need to neutralize it somehow since it won't outgass and it will greatly slow your bacteria down while you are trying to establish. If you are on well water, disregard but it will be a good idea to practice with your test kit on a sample of your water just to see what levels you start out with.
While it is possible to have more filtration than you need, I don't think there is really a situation where having too much filtration would hurt anything. The truth is, if the grow bed on that system holds 10 gallons of expanded clay balls and is flood and drain, you should be fine for 10-20 goldfish. However, if you feed them enough to get lots of plant growth, they will outgrow a 10 gallon aquarium eventually. It won't hurt you to have extra filtration but I don't know if that set up will lend itself easily to adding an additional under gravel filter without needing extra pumps and things.
As to aeration and having too much. The only drawback to too much bubbles is if you have the water roiling so violently that the fish become totally exausted by trying to keep themselves upright. So, Bubbles good, boiling the goldfish right out of the water bad. With simple air pumps and air stones, you can't super saturate the water so the limit is simply allowing the fish some calm space so they can rest.
If you already have fish in the tank, don't pee in the system, you can cycle up using goldfish. If you don't already have fish in the system, sure pee in it, it is cheaper and just as fast as buying bacteria in a bottle. But don't pee too much in it since it is a small system and a little goes a long way. It will take several days for the urea to convert to ammonia so it won't show up on the test right away, don't be tempted to dose again too soon as the delayed reaction tends to cause overdosing and too high an ammonia level will slow the whole process. You only need to bump ammonia up to 1 or 2 right at first then wait.
Do you have your test kit yet? You will need to test for Ammonia, Nitrite, Nitrate, pH and temperature. A freshwater master test kit (the kind with the test tubes and drops) is what most of us here use and they work well. Test strips are usually not too accurate.
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I intend to grow plants in the grow bed, float some on the aquarium's surface, and plant some in the aquarium. For this I was told I need a substrate. Is that necessary? If it is, should I mix some in with the expanded clay in the growbed?
Substrate? I'm not sure what you mean. The clay balls in the grow bed is like a substrate and plants grow fine directly in the clay balls or gravel. You will probably not have enough nutrients right away to support a planted aquarium as well as plants in the grow bed. However, as the system becomes more established, extra plants growing in the aquarium might be good to have.
good luck with it, I don't think anyone else on here has gotten one of those systems yet. Please post pictures once you get it set up.
Backyard Aquaponics
) ... look forward to seeing your next real AP system...