People fuss so much about the nitrite levels in an aquarium, but cycling involves a temporary spike in nitrites. Yes, nitrites are hazardous to fish... this is part of why illness, stress, and losses. This is a well-established truth to the fish-keeping hobby. But the debate as to whether or not to do water changes, I would think, is an easy one. Don't.
My reasoning? Think of your filter as a tiny ecosystem. There is food and those who must eat said food and then there is the waste they create. Ammonia is the food initially. A set of bacteria consume this ammonia as a source of energy and convert it to nitrites. But the nitrites are food to a second set of bacteria who consume it for energy and produce nitrates as waste.
So... we have a bare tank that has just had water poured in. Either fish (who create ammonia in their waste) or straight ammonia was added. Now there is food in the environment which attracts and builds up colonies of bacteria who thrive on this food and convert it to nitrites. How do they get in the water? Simple. If you don't introduce them on the bodies of your fish or via old tank water from an established aquarium (and some water conditioning solutions like 'Cycle' have them as an ingredient) since 3/4th of the world is covered in their habitat, they are also randomly in the air, floating dormant until they fall into water. Upon contact with moisture,they break free of their sealed pods and begin to feed and multiply. Hence why using plain conditioned water is sooo much harder to cycle. You're
literally waiting for nature to take it's course.
Anyway, the Ammonia will stay high until there's enough bacteria to balance it out and meet the tank's supply (like a herd of deer with grass, they will reproduce until there's just enough deer for the amount of grass). At this point, their waste builds up. Nitrite. Food for the next set of bacteria! They will start building up just like the ammonia-eating bacteria did. When
they are done, they excrete relatively harmless nitrates. There's nothing that consumes nitrates, so this is where the cycle ends. Normal water changes are done just to remove the accumulative nitrates, hence why there should always be trace amounts of nitrates in a tank. Because and efficient system has the ammonia being eaten and turned into nitrite, which is immediately consumed and made into nitrate.
Sounds much more simple when put that way, huh?
Now, back to the whole "I have high nitrites" thing. This is where people make the simple error of doing premature water changes. If you start taking away the nitrite-rich water, you're starving down your nitrite-eating bacteria (cutting back the grass the deer eat) This starves down their population to where they don't keep up with the amount of nitrites being produced by the normal ammonia-eating bacteria. You'll continue to struggle with the nitrite-eaters until they finally build up a colony large enough to handle all the nitrite that the ammonia-eaters make. So you might as well just let the nitrite spike and allow the bacteria to rise to the occasion. Your tank will generally cycle faster. There have to be enough 'deer' to consume all the 'grass' in order for your tank to cycle and become balanced.
This is also why adding a large amount of new fish creates a mini-cycle. Because you're adding more 'grass' to feed the 'deer'... more fish means more ammonia and more nitrite being produced than what the colony normally eats, so they have to boost their numbers to make up for the new production before it will balance again.
Terrible as it is, the best thing you can do is just leave the tank as-is and hope for the best. This is part of why fishless cycling was created. So that fish wouldn't have to endure the stresses of the fluctuating toxins before the tank was balanced and ready for them. But I've always done fish-in-tank cycling, so I'm not about to say it's evil... it just has its drawbacks like anything else, but that's just a personal opinion.
I can't solve the puking-barb dilemma... but I hope this helps to shine some light into the often-boggling reason why cycling is necessary and what happens when we cycle aquariums and why it goes through the processes that it does.