When we change the databaseconnection to "mysql:///galaxy?unix_socket=/var/run/mysqld/mysqld.sock" in the universe_wsgi.ini file and run Galaxy we receive an error like:
"Access denied for user 'root'@'localhost' (using password: NO)") None None
So is there an option somewhere which we also have to change to make the connection work?

You need to specify a username and password to access your database. Try this URL:

mysql://user:password@localhost/galaxy?unix_socket=/var/lib/mysql/mysql.sock
(exchanging 'user' and 'password' according to your settings)

And question two:
Do we need to migrate the content from the sqlite database to the MySQL database by hand or does this happen automatically?

What I would suggest is to dump the sqlite database to standard sql,
and then creating a new instance of galaxy that has mysql as the
configured database. When you run the setup and run script, you'll
have a freshly instantiated galaxy with empty mysql tables. Then you
can just import the standard sql file into the mysql database.