It's as bluebell says-- even the most heavily-stocked tank will not provide the CO2 necessary for aquatic plants (except perhaps the slowest growers). Especially in a well-aerated tank, CO2 will be lost rapidly.
If 2/3-1/2 of the ground space is planted with fast growers, you already have enough plants to have a heavily planted aquarium.
1. While it might be possible to grow the plants without CO2, it would be heavily recommended. Especially if you have high lighting, the plants will be especially hungry for a carbon source. Ie-- the higher the lighting, the more carbon the plants need, the more CO2 that should be injected. Remember too, that if well acclimated, fish can stand moderate CO2 levels because the plants will be rapidly producing oxygen as well.
2. 2/3 is fairly well planted, and even with 1/3, or mildly planted, CO2 would still be a good idea. If you plan to grow plants well, you need a source of Carbon. Most hobbyists use CO2 because it is the most effective and simplest. Excel could work, but it is more of a supplement. Walstad's non-CO2 method uses floating plants to absorb carbon from the air that will be released into the water. Bottom line is that the plants need Carbon somehow.