Based on the MySQL error you are getting, I believe one of the most common issues that may cause this error is that the system or init script is passing a different path for the socket than the one defined in your my.cnf. Thus, it returns a result of a failure when MySQL attempts to start or restart.
A possible workaround for your local, would be to check the my.cnf and all associated configuration files and evaluate what is being defined for the socket. If the MySQL configuration is defining something other than what the startup script is asking for then you can change it to the following: socket= /var/run/mysqld/mysqld.sock. Then click save edit and restart the MySQL again. Hope this resolves the MySQL issue you are encountering.
[1] Creating a SQL dump file: https://cloud.google.com/sql/docs/mysql/import-export/creating-sqldump-csv#sqldump
[2] Before you begin: https://cloud.google.com/sql/docs/mysql/import-export/importing#before_you_begin
[3] Importing SQL dump file: https://cloud.google.com/sql/docs/mysql/import-export/importing#importing_a_sql_dump_file