This would depend on your physical ability. Walking an 18-hole course would be similar to walking 6-8 miles, or hiking 6-8 miles depending on the course.
If you are really out of shape, then start with eating better and walking a lot more. Then you can ease into jogging. It might be best to look into interval training. Jog for some time, then walk, then job. Slowly go from to more jogging than walking.
If you want to make sure you don't hurt yourself, then I would ask a doctor or a trainer this question.
Depends if it actually causes an issue. You have golfers who keep their eyes looking at were the ball was longer than others, and you have golfers like Annika that start looking forward before they even hit the ball.
I am not sure it is related to eye dominance.