Please post the URL of the blog your refer to as the one linked to your username is wearing Twenty Eleven and not Twenty Twelve.
There have been issues with the theme customizer. However, you do not need to load the design tools unless you have the Custom Design upgrade, which it appears that you have not purchased. All of the very same functions can be accessed from your dashboard.
Appearance > Header
Appearance > Background
Appearance > Theme Options (applicable only to some themes)
Settings > Reading (set a static front page)
Site Title and Tagline > Settings > General
If you want to change themes on a free hosted WordPress.com blog you must be logged in under the exact same username that registered the blog and who is the Admin of the blog. Note that no content is ever lost when changing themes.
Go to Dashboard > Appearance > Themes
browse themes until you find one you like and click "activate"
or type in the name of the theme you want to use and
click the "activate" link when it appears
After you change your theme all you need to do is go here Dashboard > Appearance > Widgets > Inactive Widgets and re-install them where you want them to appear. The widget contents and settings will be the same as they were prior to changing themes.