s_babb, when you change to a new tool, it turns off symmetry. It actually is like starting fresh each time you change to a new tool. So, let me see if I can describe this to help you.
In the Tool Palette where you have golden “Z”, the large one, not the small one… each time you switch that out you are selecting a new tool and starting fresh. Some of your settings may persist like materials and the current color, but as far as a lot of the toggle buttons and sliders found in many of your palettes they go back to default.
The rest of the buttons which near the larger “Z”, which are smaller, those are tools which you have worked with. The tools you worked with retain the settings where you last left off. So if symmetry was on for a tool and you leave it to work on different tool and come back, symmetry still will be on. So again, if you select from the larger “Z” you are selecting a new tool and it presents the tool with it defaults.
Also when you select a new tool, like 3D cylinder, sphere, the star etc, it may start you in edit mode automatically if you have been working on previous tools during your session, but you must make it a polymesh to sculpt on it. Again this is only when you are picking a new tool, not a tool you have been working, which are the smaller buttons.
I empathize with the frustration and I encourage you to stick out with zBrush. I truly believe, and no offense by the way, that what may appear to be broken features is really user error. I think one has to get accustomed to how the software works because if you look at many of the users here who are power users and submitting their art work for show and tell, they do not experience what you are going through. I experienced some of the frustration you are experiencing the first week of using zBrush getting acquainted with it and masking was just killing me because I had no idea that I was applying masks or forgot to clear one. So I when I went to do something and there was mask, the feature or action I wanted to do on the model would not work.
I been studying zBrush taking classes with Digital Tutors and that has reduced a lot of my early frustration and shorten my learning curve. You learn a lot if you continue to plug away at the course material. Anyway again, I encourage you stick it out and be persistent.