Cannot find mysql.sock

to answer the first part of your question:

run

% mysqladmin -p -u <user-name> variables

and check the 'socket' variable

This solved my problem

mysql_config --socket

UPDATE

mysql_config can tell us where the file mysql.sock should be, but in my case the file didn't exist. So, I've deleted my.cnf:

sudo rm -rf /etc/my.cnf

And then restarted mysql:

brew services restart mysql

The file was created and mysql is now running well.