Glad to hear the nipping behavior has improved. I hate to say this so directly, but I am pretty certain there are some issues with the tank as it is that you may wish to address. I'm glad to hear you are considering a larger tank, because 3 gallons is too small of a tank to support 8 fish, each of which could potentially grow to 1.5-2 inches. One of the issues is waste dilution-- 3 gallons unfortunately isn't enough volume to keep nitrogenous waste at a safe level with the amount of waste these fish will produce. Another related issue is that it will be hard to keep water conditions stable in a tank that size-- these might range from abrupt temperature swings, changes in pH, ammonia and nitrite spikes.
One other issue I might point out is that cories don't tend to be terribly good fish to use to cycle as they are rather sensitive to nitrogen.
I don't have any experience with the Eclipse line of tanks, but my understanding is that the filter systems aren't terribly strong or efficient; perhaps someone with more experience than me could corroborate or denounce that.
I'm quite convinced you and your fish will be happier with a larger tank. Bigger isn't always better, but with aquariums it tends to be

Most people recommend 20 gallons or larger for your first tank, but if you can't make the space then 12 is certainly better than 3.
It sounds like you have done some research so far, which is great! Also, doing daily water changes of 20% is a good intervention. I'd recommend returning the cories or placing them with a friend who has an already cycled tank until you have either upgraded your tank or completed cycling the 3g. To be honest, I'd recommend doing the same thing with the tetras. Then, you can complete the cycle in the 3g and keep just the barbs and possibly a pygmy cory or two, or upgrade to a 12g or larger and get all your fish back
It's hard to give such hard-line advice, but my feeling is things will work out better in the long run with this approach. Hope that's more helpful than not~
*Edit: I forgot to address your question directly! Waiting for ammonia and nitrite spikes is often the bane of a fishkeepers existence. The cycle as a whole depending on what method one uses can take any where from a couple of weeks to 6 weeks. From what I remember nitrites show up when ammonia starts declining (as the first bacterial colonies grow in). Nitrogen-metabolizing bacteria are slow growers, so this can take 2-3 weeks to start happening, but your mileage may vary.