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It is a good idea to clean the gravel when changing water. An air stone will keep the maximum amount of O2 in the water as possible. The bacteria like well oxygenated water.
You don't need to shift the gravel as much as just vacuum up fish waste and other excess tank material.
Air-stones help to agitate the surface promoting gas exchange but other than that doesn't really have an effect on cycling ... other than some more oxygen.
The only thing I saw in the article that I would do differently is that when you re-use the filter sponge or cartridge it says to rinse with a strong spray of water. To me that suggests rinsing it in the sink. I think it's best to rinse it in the bucket of "dirty" aquarium water from your water change. I use a cartridge so I just brush off both sides. I've been using the same cartridge for 3 months. I think most people keep using them until they kinda fall apart.
Air does help the cycle, the nitrifying bacteria that convert ammonia to nitrite to nitrate are aerobic bacteria.
sorry. didn't know it was fish in. just vacuum random spots each time. If you can see waste then suck it up, but don't vacuum the whole floor every time.