Whenever I turn off my monitor for, let's say, more than 3 hours, it displays what appear to be 10-15 lines running across monitor in a diagonal fashion -- what problem is this indicative of? The problem starts go away after 5 minutes. I didn't have this problem when I bought my monitor 2 years ago.
Is my monitor dying? If so, how long do I have before it craps out?

This is a common problem with your Sony G1 chassis. What you are seeing is retrace lines. If it has not got too bad, you can correct it by running the 'Image Restoration' option which you will find in the Color menus of the On-Screen Menus. For this option to be available you must have the monitor powered on for a minimum of 30 minutes without going into power save mode.