Has minimally invasive surgery replaced open surgery in gynecology?

Not yet. Not yet but that's the goal. The advantages of minimally invasive surgery is incredible, but there is a high learning curve to master which makes it a little more challenging to make this into the gold standard. We are making great strides however.
To a certain degree. As technology and medicine advance together, better instrumentation, better medications and innovations in minimally invasive surgery allow for most benign and some cancer gyn surgeries to be performed this way. In our center we perform nearly 99% of all of our surgeries in a minimally invasive way. Less scarring, less bleeding, less pain and shorter recovery. Less is better!
Laparoscopy. Not necessarily - the treatment needs to fit the diagnosis, the patient, and the expertise of the physician. Laparoscopy is a very useful technique or tool.
Not necessarily. One type of surgery has not replaced the standard, traditional, and well respected open surgical procedures. Laparoscopic surgery is one good option for some patients in gynecology surgery, depending on the patient's condition and the surgeon's expertise.
Not completely. There are minimally invasive gynecological surgeries available, including those done with the DaVinci robot. That being said, there are still some conditions that are better done through the traditional open surgical technique. I would discuss the specifics of your case with your surgeon.
In many ways yes. Many gun procedures can now be done minimally invasively and have or should replace many open surgery techniques.