After the infected stage, the parasite leaves the body of the fish and goes to the substrate to reproduce. Then the new parasites go free swimming, and its only in the free swimming stage that medication is effective.
Note that different individual parasites will be at different stages of their lifecycles. Not all will be free swimming at the same time for instance. Some parasites will be infecting fish, some will be reproducing, some will be freeswimming.
Also, healthy fish with good immune systems can be infected and show no symptoms. If not fully killed off, your tank will remain infected and you might see no signs. It could be years later when a fish is otherwise unhealthy or stressed before ich symptoms reappear.
Yes, you need to remove the carbon from the filter.
Some info on carbon.
Carbon removes organic compounds and some metals from the water. These organic compounds can be tannins from driftwood which cause tea like colour in the water, so carbon is often used in new tanks to get the water clear until the tannins have leached out. Organic compounds can cause odours so carbon is used to remove these odours. After a course of medication you want to remove the medication from the tank and carbon will do this. If you arent dealing with any of these issues, carbon is doing nothing useful. Tannins are temporary and should leach out of driftwood after a short period of being submerged. If you keep up with your tank maintenance, there shouldn't be smells. Medication is only a short term occurance. Ordinarily you dont need carbon, it isnt doing anything. Carbon only lasts a few weeks before it needs replacing, its expensive. You really dont need it in your filtration. Use it for its intended purpose, then stop using it once whatever you are dealing with is done.
As to your other filter media, yes leave that in there. But be aware that the active ingredient in ich x is malachite green, which is a mild anti biotic and might kill off the microbes responsible for your cycle. Malachite green is also a green dye, and it will stain silicone, tubing, rocks, decorations, your clothes, etc.