If my histogram spikes in the middle and evens out pretty much evenly but not completely to the ends (0 & 255), does this just mean that my scene doesn't have the deep shadows or brightest highlights to register on the histogram, and therefore does not reach the complete dynamic range on my camera?
Pretty much, yes. A histogram that looks like that can represent a proper exposure or an exposure that's off a bit, but it can't arise as the result of a gross mistake.
If, for instance, you had something that was as absolutely black as it's possible for something to be in nature, and you had metered so wrongly that the black value was sitting in the middle of the histogram, then you'd expect anything in the picture that wasn't black to fall off the right side of the chart. The same sort of thing would happen if you exposed for a light source, but in the opposite direction.
It is common (and usually desirable) for studio shots to lie completely (or almost completely) within the camera's dynamic range, with perhaps just a handful of pixels representing a true black or white. (That assumes, of course, that you haven't tried to create a digitally white or black background by overexposing the background or using something like black velvet.) And Ansel Adams would have considered it a great personal failure if anything important in his images fell below Zone III (two stops below mid grey) or above Zone VIII (three stops above).
By a lot of standards, a histogram that looks like the one you've described is an ideal exposure of an ideal scene. The only reason to worry about it is if the image appears too flat. If it's an image you're creating in studio (or at least one you're lighting yourself), you can make lighting adjustments to make the contrast more appealing. If it's a natural-light shot, you can always increase the contrast in post. When the histogram hugs the edges, you're kind of stuck with manipulating small parts of the contrast curve, and if it actually hits the edges hard, you're kind of stuck with what you've got.
Yes, the dynamic range of the camera is shown by the histogram, and if your image isn't spread across the whole range of the histogram the photo doesn't take advantage of the full dynamic range your camera offers.
This is not necessarily a bad thing (or a good thing). It's important to remember that what the histogram shows is a representation of the image: it's neither good nor bad. It's up to you to interpret that histogram to decide if you need to do something more during capture or post, or to just be satisfied with it.
For arguments sake, let's say you wanted to take a photo of... oh, I don't know, a person. For whatever reason you decide to take a close-up of their cheek, their skin. Click goes the shutter; look at the histogram. It doesn't show values stretching the full width of the histogram, correct? Remember, stretching the full width means the full range is encompassed in the photo, all the way from 0 to 255 (black to white, for arguments sake). Is there black in the skin? Is there white in the skin? Of course not, it's much more of a mid-tone, and only a mid-tone. Look at your photo -- does it capture that? Yep. So even though the camera did not record an image that made use of the full dynamic range it's capable of, you still took a perfectly good photo that adequately captured what you saw.
Possibly. One thing to consider is that the kind of histogram you are talking about, with just grey values from 0 to 255, is only an average of the color channels.
It's very possible that (for instance) the scene could have a lot of deep blue and bright red elements - the blue could extend all the way to dark shadows and the red could extend out as far as your camera could go, but the histogram would look like it was not near the edges on either side.
That's why RGB histograms are a lot more useful, because you can see when a color channel might be clipping on either end.