In the theme for one of my websites, with ai1ec in month display mode, neither the display selection button nor the categories button work. I tested ai1ec with the WP default template and everything worked fine, which tells me that my host and ai1ec are not the problem.
It dawned on me that there are a number of components to a website that have to work together: 1) the theme, 2) ai1ec, 3) the other plugins that are installed and, 4) the host. Any of these components can work 100% by itself, but conflict with any number of other components.
To my way of thinking, the very first step in resolving a conflict is to test ai1ec with the default WP theme. If it works, you know the host, ai1ec and your host are playing well together. Then, you have to deactivate all your other plugins and see what happens. Activate the other plugins one at a time and see if your conflict reoccurs. Eventually you will (or should) find what is causing your problem.
Just my thoughts.