Do you suggest laparoscopic surgery for ovarian cyst prior to open surgery?

Yes. If a surgery is amendable to laparoscopic, then I always recommend going that route. The recovery is faster, the pain is less, lower rates of infections. Not all surgeries can be done laparoscopically, but if it can be then do it.
Yes. Most times cyst can be taken care of laparoscopically by an experianced gynocologist. Lap surgery generally leads to faster recovery, less pain and safer outcomes when compared to open procedures. If your given a choice of a laparoscopic procedure vs and open one its worth it to try the laparoscopic approach. Also laparoscopic surgery can always be converted to and open surgery if needed.
Absolutely. In addition, robotic surgery has opened the door to much less invasive surgery for complicated surgery that traditionally has only been done as open abdominal surgery.