First select and 'ungroup' the audio and video tracks using the broken link icon. You can then do what you like to the separate parts. Depending on what you going to do, it might be wise to select the remaining parts and re-group them so the audio and video bits remain in sync.
Note also that when you make a cut you've done nothing to the actual file, you're just making internal markers for what parts are to be visible and where on the timeline. A 'cut' object can be restored by simply dragging out its cut edge.
Remember one thing: It is not always necessary to ungroup and delete. The "direction of view" is from down and up. So if you have the original video and audio on track 1 and 2, and you place a new picture or video (without sound) on track 3, the video on track 1 will be in the "shade", and not be seen. You now have a new "background" on the audio track. You need not remove the video clip on track 1.