The carbon is just to help remove tannins. When the driftwood tapers to almost none released, then you can remove the carbon, as it's really not needed. The only 2 real uses for carbon is removing tannins and removing meds if you had to medicate the tank. Other than that, doesn't do much anything else.
Tannins are tannin acids, which wood releases. Not harmful at all to fish. Just might temporarily discolor the water, but will eventually go away over time.
Is the other tank big enough to put the filter from the main tank or too small? Could just use your current filter if the tank isn't too small. And also use some of the current gravel in the bottom, which will help preventing a new cycle and will keep your filter cycled.
Air stones,
IMO, is just up to the individual. Some like the look, and others don't. I don't use them myself.
One last thing, once you begin the change, should be able to do it in a couple hours. Took me just over 2 hours to get the sand rinsed, added, and tank filled, then a 75%
pwc to help get rid of cloudyness. Then 2 hours later, I added my filter, as the water was crystal clear. Then it was just adding the ornaments, etc, and I was done. Then I did a fishless cycle before adding fish. But since you already have a cycled filter, that won't be necessary. So the running of the filter on the other tank will be to just keep it cycled and the bacteria alive, along with the goldies.
Also, if you have an empty spare filter bag, you can put some of the current gravel in it and leave it sit on the bottom of the new tank setup to help bacteria spread in the sand as well.