I have tried utilizing a simple form to get the hang of how your scripts work, but am confused. I copied the section of "Sign-In Method" from the "Account->Settings.html" page in demo 20. By clicking the "Change Email" or "Reset Password" buttons, neither of the sub-sections open up and I was curious to know if there is something I should be looking for. I copied it exactly from the settings.html file, so I haven't changed anything necessarily to fit my project, I just wanted to see it in action.
I would suggest you to double-check your code and compare it with the original theme's template. Most probably you are missing the page's js script that handles the form settings sections so you should double-check your console for any possible js errors.