You’re asking two different questions here. Certainly a priest can be close friends with a woman. But, no, he should not give the appearance of dating her. The activities you suggest would constitute serious scandal and a near occasion of sin, at the very least.
Priests must be very careful to be circumspect in their relationships with the opposite sex. I can only recommend that any friendly get-togethers be with groups of men and women and not one-to-one encounters. (For example, before he became pope, John Paul II often went hiking and kayaking with groups of lay friends, often married couples whom he had known and counseled since their school days.)
