My questions are whether there is ANY advantage to 8MP vs 5MP with the DX
Not really. There really isn't the resolution of the lens, or low enough noise, to take advantage of 8mp or to make big prints, and 5mp is more than enough to share photos electronically or with small prints.
what I'm doing wrong when the camera gives the normal(?) 2 beeps but no picture is taken
Two beeps indicates focus, not a photo. Either you're not depressing the button far enough, or your button has problems.
but no picture is taken, and why 2 pictures taken minutes (or less) apart differ SIGNIFICANTLY from one another.
Metering, mostly. The metering on these cameras is pretty much shit. A good camera app would likely help, and I haven't quite pinned down the metering habits of the stock camera. What I do know, is it tends to give weight to the focus point when metering, but even a small amount of bright area in the photo often causes the metering to go batty trying to compensate.
For example, I was at a lake the other day taking a few shots of the ducks in the grass. If I had any part of the lake in the shot, the reflecting light was causing the photo to be well under exposed. Take the lake out, and boom, no problems.
The meter should not be so easily overwhelmed by wide variations in light - if 80% of your scene is dark, it should lean towards exposing for the 80%, not going loony to capture the 20%.
Does it let me know that it cannot focus? How?
I believe the focus bracket turns green when it believes it has focus, and stays red when it doesn't. However, focus is not to be trusted blindly, always look because no focus system is perfect.
Is lack of focus ability part of my issue? I often try to take a snapshot of a machine part label where usually the closer the better.
The camera doesn't like to focus close unless you turn on macro mode. At least, in my experience.
On the last, I've kinda decided that the "shutter" fires at different times, and my really blurred images come when I move after I think the photo was taken and it wasn't. Perhaps enabling the shutter sound will help me there.
There's a visual cue on the shutter firing. The picture gets frozen and there's an animation of it moving to the lower left corner if memory serves me correctly (sorry, I left my phone at home). Go with that.