Your most likely having an issue with the plugins. These types of problems are extremely hard to diagnose. I had a similar problem with the Wordpress theme Arjuna-X where it was loading it's own modified JQuery library that was only breaking the switching of "Visual" and "HTML" in the post editor. Took me days to figure out.
Essentially go to Plugins in your wordpress admin. Then next to the word "Plugins" there is a little check box that when clicked will select all your plugins. Then use the drop down to select the "Deactivate" option. Then click Apply.
Now go check if the problem is still there. If it is then I'd suggest re-installing the theme (I know you did already, I'm just putting it here for reference). But maybe also try replacing the physical theme directory, not just re-installing the them from within Wordpress
Then one by one enable each plugin and take note of which plugin breaks the widgets section. Once you found the plugin, just search the internet for that plugin name, something like "[plugin name] breaks widgets area". Also you might as well email the author of the plugin or check to see if the author has a temporary fix. It could be as simple as a setting or you might have to leave that plugin deactivated until the author fixes the problem. Tell us if you come right and what plugin was the issue.