Well Milton Sand, I know I'm a little late, but maybe I can still help you with this.
1. When do you use "to fall over" instead of "to fall down"?
2. Why your second suggestion doesn't reads "falling down"?
1. When using "fall over" it generally means that that person is sitting in a chair or falls to the ground, but tipping over. It can also be from a standing position when someone faints or goes from one position and falls without being able to break their fall. I hope I'm explaining you this good enough.
2. His/her second suggestion could also have "down" added as in "She is always falling down." either is correct.