Cant start mysql and cant find mysql.sock
Checks: 


Is the server actually running locally? 
Grep for mysql.sock in my.cnf and provide results
Run: sudo /usr/sbin/lsof  | grep mysql.sock
Check (i.e. grep) mysqld err logs to see if its remarking about mysql.sock or any other startup issues.
Check permissions on the MySQL directories, i.e. /var/lib/mysql/ should be read/writable by the mysql group and sufficient permissions for users to reach the socket file. 
How are you starting the mysqld process? The error looks more like a client connection issue? 


While you can re-install solutions I have used in the past include: 


Touch the socket file and make sure it is owned by the mysql group and user. 

ls -l /tmp/mysql.sock
srwxrwxrwx 1 mysql mysql 0 Jun 15 15:08 /tmp/mysql.sock


Check the my.cnf for the setting for where the socket is.

You can use the following steps to re-create the socket. Just change the name to reflect the expected name of the socket as given by my.cnf under the [mysqld] and [client] sections.

sudo touch /tmp/mysql.sock
sudo chown mysql:mysql /tmp/mysql.sock
sudo chmod 0777 /tmp/mysql.sock

Starting mysqld as a service via systemctl mysql start etc.
Make sure you are using sudo etc. as most command require root level privileges to work correctly.