When I test your files I'm able to add items to the cart without issue so it seems to me that there might be a session management issue on your server. To help you test the sessions more thoroughly you should add in this code on the pages you are testing so that you can see exactly what is in the session during each step.
This code will print out the entire session so that you can see all of the values in it. You should add this to your pages then test the site locally and test it again on the live server. Let us know what the differences are between the servers in the results that you get.