I just saw this tutorial on how to do refractograph. The tutorial specify that you have to remove the lens to make this work. Do you think it would be possible to do something similar with a lens on? If the answer is yes, then which type of lens?
Yes, you can do it with a lens on the camera. The lens will just become apart of the refracting object and introduce a "focal length" into the picture. The plane you are capturing on is the camera sensor (or film) and not something else.
So it will work with a lens on but you wont get results like in the tutorial due to the additional object and "zoom"/focal length in the light path. If you are concerned about dust in your camera you can use a cpl mounted on the lens mount.
No, or at least not directly, and not with the clarity and fine detail that you're seeing in the example images.
The image that is formed by a refractograph is the caustics of the light directly impinging on the sensor. You're probably more familiar with caustics from underwater pictures and film, where there are patterns of variation in the light falling on the subjects being photographed. With a refrectograph, your sensor (or film) is the object upon which those patterns are being cast. In a sense, the object you are taking the refractograph of is the lens.
In order to capture those patters using a focusing lens (whether that lens in in focus or not), they need to be falling on something that you can shoot. Whether you are shooting at a reflective opaque screen (a piece of very smooth, very reflective coated white paper) or through a transmissive screen (such as a one-side-matte mylar or acetate film or a ground glass focusing screen from an old camera) the screen itself will limit the dynamic range available to you, and probably add a bit of its own texture and reduce the detail quite a bit.