mysqld startup/initialize permission demied - can't connect
It looks like permissions for your actual data/* is correct: mysql:mysql, but if you're having it access that through a symlink in your own user's $HOME, then the permissions leading up to the symlink are probably what is blocking mysql from accessing the symlink path. 

What about giving mysql its own $HOME, like /home/mysql/{etc,data} (chown mysql:mysql) and place its configuration and data files within that. You could always add your user to the mysql group if you need to access mysql's config or data as your own user.