1. Only thing I can think of is to extend one clip past the out point to restore what was cut out
2. If you delete the audio track from a clip it is gone. Based upon the clip’s time code you should be able to go back to the raw footage in the bin and find the same shot.
You can copy and paste a video only clip into a new sequence, export an EDL, open the EDL in a text editor and add the Audio 1&2 using the correct format, then batch digitize that sequence or a new project, which will restore the same clip/s with audio. This will take longer than manually finding the same clip with audio from your original material in your bin. But I have done this for a whole sequence which I somehow captured sans audio.
3. I do not know how to skip cuts on selected tracks when using page down to go to the next cut. If you have recently had a good cup of coffee, however, it should be easy enough to eyeball it even at a rapid pace. Also, if you are like me and do Page-Down + Ctrl-D repeatedly, you needn’t worry about the cuts on non-targeted tracks, so just go for it. You may get some RSI from the extra keystrokes, but at least you do not need to use the mouse.
4. You can assign a keyboard shortcut for cut. I use Ctrl-/ since this is what the Media 100 uses and I worked on that system for years. Look at the keyboard customization menu, you may learn some new features.