With no other information, all four would mean that you got to know her on one occasion. If you added something like "every day," then all four would mean that you got to know her over the course of many meetings.
"While" is better than "when," but both make sense.
In this situation, we would usually say "woman" rather than "lady," which is old-fashioned and can have odd connotations.
Who "her" refers to is slightly unclear. Picking up your daughter has a very different meaning than picking up a woman. We know that you probably mean you were picking up your daughter, but the sentence can be understood either way. So it would be better to say "while I waited to pick up my daughter at school."