I can go to the Windows menu, scroll down to "Text" and then select "Character" and it will check and uncheck each time I do that. Simultaneously, each time I check and uncheck "Character," the Character menu in the right studio will appear and disappear. Which is what I would expect it to do.
After closer inspection, I found that if I click the "a" button when the Character menu is not open in the right studio, the character menu will then appear in the right studio. But if I click the "a" button again, the character menu does not disappear. I kind of figured it would, but if it's not supposed to then I guess I'm good.
Also, to address what Walt said, I was expecting a free floating character panel to appear when I clicked on the "a" button. So you were absolutely correct in that regard.
Once again, thanks to all for your assistance!