What do you think about exercising early in the morning, immediately after you woke up? Isn't that bad for the organism?

No. Many people exercise early in the morning. There is no harm to it.
Good warm-up. Exercising early in the morning is preferred for many people because of their time schedule. It is fine as long as you are alert and have a proper warm-up. A proper warm-up means beginning your exercises slowly and at a lesser intensity than you would 5-10 minutes later in the workout. Doing cardio (walk, bike, elliptical, slow jog) for 5-10 minutes before you get deeper into your workout is wise.