Is it likely for a person to experience hair loss after just one round of chemotherapy, or does it take more than one treatment? Is there a chemo drug with less harsh side effects for breast cancer

Depends. This depends on the strength and the type of therapy that you receive. These questions are best answered by your oncologist.
NO. Chemotherapy is given to kill cancer cell and save the person, not to save the hair, some will loose after initiation of treatment hair will grow back in 2 to 3 wks after completion in most of the time , in very few the hair roots are killed will never grow back.