Having Trouble with Autofocus

Still, even if that is the solution, to me it kind of begs the question of WHY the camera is so often not in focus even though Autofocus is turned on. And then the next question, does that mean that the image truly is out of focus, or that the camera just thinks it is for some reason?

Hi, I’ll try to answer your questions. First it could be something wrong with your camera or maybe a dirty af module but I wouldn’t think so. Instead it probably has to do with the circumstances under which you shoot. Like to dark, not enough contrast where the focus point is on the subject or a lens that focuses very slowly so you have to wait a little bit longer before triggering the release button. 
Try with different focus modes (single point, group mode, dynamic mode) and see if you get better success with one or the other. I wouldn’t recommend the af-a mode which often is slow to find what it thinks is the best way to auto focus for any given situation. If I remember correctly there is a description in the manual about af-a where it says something along those lines.
Your second question is partly answered above but I can add that at no time you can get the camera to do more than “think” weather that is in focus or out of focus. In really tricky situations one has to rely on focusing manually. Luckily for most of us casual shooters those occasions are rare.