Re: cycling tank - seachem stability
Keeping the ammonia bellow 0.25 ppm should be fine, the main thing you are trying to do is protect them fish while cycling. That is why using Prime every 48 hours will be so helpful. It will keep ammonia, nitrite, and nitrate detoxified until the beneficial bacteria can consume them.
As for the carbon, removing bacterial cultures is exactly why I would leave it in for the time being. Once you see no ammonia and either low to no nitrite but your nitrates are rising you can add some fresh carbon.