The purpose of Liveview is for focusing and framing, just the same as the optical viewfinder. Of the parameters for an image - ISO, shutter speed and aperture, none of them are applied to either view. The camera, especially in low light (night, indoors, etc.) situations essentially gathers light in order to present an image in Liveview that is useable. You might notice at night that there is a very substantial lag in the view of the rear monitor, while the sensor collects enough light to put up an image. You can see something move with your eye, then a second or so later see it move on Liveview. That lag is the amount of time the sensor needs to gather the light so that the image processor has sufficient information to form a useable image on the rear monitor.
In terms of the double-shutter, lets walk through the operations. You turn the camera body on, then put it into light view mode. This will lift the mirror and open the shutter, thereby letting the light in to the sensor and the Liveview image appears. When you depress the shutter, what happens is that - the camera is still an SLR - so the shutter slides back over the sensor, and then activates. The shutter is a dual curtain - a forward curtain followed by a space and a rear curtain. The space is calibrated for the shutter speed actually being used - between 1/8000 of a second to 30 seconds. Then after the image is captured with the shutter sliding over the sensor to gate the light in, the shutter then is slid back opening up the sensor to the light and thereby returning to Liveview mode. This is a bit different than that of a P&S or a mirrorless (like the Q), where there is an electronic shutter on the sensor. You can think of this function as the image on the sensor is cleaned off, then the sensor is commanded to accumulate the light for the period of the shutter speed (lets say it 1/30 of a second), then the sensor sends this information to the image processor for processing, while the sensor is returned to providing the scene back to the rear monitor.
You can put the camera body in live view, flip the body around and take off the lens - and look into the camera. Then press the shutter and you can see the shutter roll back across the sensor, then the curtains with the slit - and then the sensor slides back off the sensor opening up to return to liveview.