Yes, you can control robot arms and move objects around in simulation. You will need to setup the simulation environment, get models for your robot arms, and write some code to move the arms around in your desired trajectories.

Start with the tutorials.

If you don't care about physics and you just want a video, then you might find it easier to use something like Unreal or Unity3d.

Showing the results of subtractive manufacturing such as milling or drilling in real time, is a non-trivial task. The V-REP video you linked to was just carving the shell of the original work piece not doing a full 3D carve. You may be able to find a good open source constructive solid geometry (CSG) library that will allow you to load a mesh and then carve it with a cylinder, to simulate milling. The challenge then will be update a model in gazebo in real time, which will require a Gazebo plugin.

In short what you want to do is possible, but is a long way from an easy task. I'd start will building a model plugin that allows a loaded 3D model to dynamically change. Then when that's working look at adding a CSG library to plugin so that another link from the Gazebo world can be used as a carving tool. It's likely that such a system will run significantly slower than real time.

Hope this points you in the right direction.