It's true that you do not have to do a traditional cycle if you have enough fast growing plants, but the key here is fast growing. Most stem plants (but not all) and more especially floating plants are your fast growers. Plants use Ammonia as their main source of Nitrogen (they actually prefer Ammonium and will convert Ammonia to Ammonium). That is why you do not have to 'cycle' with bacteria, the plants use the Ammonia before the bacteria could have a chance at it.
However, even with using this method you have to go slow and steady. I assume you are only putting a betta in, so that may be fine, but don't add anything else for at least 2 weeks (snails, shrimp, etc). What size tank? If this is something really small, you still may have problems. Once you add your Betta, test for Ammonia and Nitrite daily. You will probably see at least 0.25 ppm Ammonia but if it goes beyond that you'll need to do a water change. If you see anything for Nitrites, do a water change. Using a water conditioner like Prime will allow you to go 2 days between water changes if you have ammonia/nitrite problems.
For your filter, yes remove the carbon. All it is doing is removing nutrients for the plants. Speaking of nutrients, do you have a plant fertalizer? If not, pick up Florish Comprehensive (check the label, Seachem has many products under the Florish line). You will need to dose that once a week, a day after water changes, to ensure the plants have enough nutrients to continue growing (and using Ammonia).