Effects of exercise on the brain
Although current research has little to say about the effects of exercise on personality, there has been recent correlations demonstrated between personality and the "drive" to exercise--specifically, aggression as a personality trait has been linked to higher metabolic rate. See http://phys.org/news206006380.html for more information about this.

The causation in exercise-induced neurogenesis is not well understood. A correlation that has been found (in mice) is the presence of a hormone called beta-endorphin, which is attributed to positive affect (a boost in mood). Exercising mice that demonstrated the presence of beta-endorphin in their hippocampal region did experience exercise-induced neurogenesis, while those that naturally lacked beta-endorphin did not experience neurogenesis. 

The basic worth of circulation should not be overlooked, either. Anything that gets your blood pumping more efficiently means that the more distal parts of the body will appreciate an influx of nutrients. A smoker with poor circulation (as a result of lowered oxygen in the blood -- carbon monoxide competes for hemoglobin-binding) will have cold hands and brittle fingernails, and this is just a simple example of this phenomenon. The value of increased circulation to the brain shouldn't be overlooked -- the blood is our main transportation system, and the brain gets nothing that is not delivered via circulation. It's akin to watering a plant -- sure, it may grow it you give it 30 minutes of sunlight and 1/4 cup of water a day, but it will certainly grow to its full potential if it is fed as much as it needs.

Exercise is important, but it's easier to think about the questions you pose if you consider exercise more as a physiological stressor on the body. Muscle tissues are torn and damaged by exercise, but the body recognizes that it must need these resources (because of some present danger in the environment that caused you to run/strain yourself) so you begin to see the typical skeletal muscle hypertrophy. The heart is also made stronger through this strain, which is why regular exercisers have a lower resting heart rate -- it takes their heart less effort to pump the same amount of blood.