1. Is it possible to create a customized directory (such as "MyMiniPlayer") so as to preserve the original Root of Panel?
You could create a custom directory in ...\Data\Panel with a customized Panel, but Media Center is going to use what's in the 'Root' folder.
2. Is there a way to save my version so it won't be overwritten by updates to MC?
Right now, Panel doesn't support custom CSS, so whenever Media Center updates, it's possible to lose any changes you've made when it updates. You could create a backup of the changes and just paste them back into place. Unless there are changes to Panel, the CSS and JavaScript mapping names won't change.
It's possible that in the future Panel might support custom CSS schemes, but I don't know when or if that will happen.