Does a high level of daily physical activity require carbohydrate intake? What about just eating protein?

Must have carbs. Carbohydrates are essential for energy. There are arguments over how many and exactly what type, but generally speaking, a moderate amount of complex carbohydrates should be a part of at least a couple of your meals each day. Those that try to exercise without carbs in their diet often have shortened or poor workouts.
Yes. Carbohydrates are not an essential part of the human diet because glucose is a breakdown product of protein metabolism. However, simple and complex carbohydrates are a plentiful source of energy and are useful to people engaged in high level physical activity. Avoid processed sugars and flour, which are nutritionally poor sources of energy. Use whole fruits, vegetables, nuts, beans, and grains.
Yes but. A good balance is very important, but with increased physical activity, additional carbs and protein are necessary. If in doubt, consult a personal trainer or nutritionist.