Ok, have reviewed your install. The findings are not good. I've no idea what did it, but all CB tables are broken due to missing their primary keys. More than likely core Joomla tables are in the same shape as well. The only thing I can think of that could do this is a bad backup tool used to restore the database, manually altering the database, or maybe you tried to copy the tables over but didn't copy the primary key and auto increment.
This is FAR outside the scope of our support, BUT I have repaired your database. Please be EXTREMELY careful when dealing with your database as at some point either you, another admin, or some tool you've used has completely destroyed the table structures of CB.
I did not examine other tables so you may still have issues with other extensions or core Joomla tables.