Cannot configure nor start MySQL

You installed mysql-server-core-5.6, which partially failed or partially installed.

The -explicit_defaults_for_timestamp is only for MySQL 5.6. Its probable that the partial installation of mysql-server-core-5.6 added this option. I am going to suggest now that you remove any trace of mysql-server-core-5.6 and reinitialize the data directory.

1. kill any running mysqld processes:

   ps aux | grep mysql
   kill pid
   

2. Uninstall the mysql-server-core-5.6 packages:

   apt-get remove mysql-server-core-5.6
   

   A list of files is here

3. Reinitialize the database directory:

   A. rm -Rf /var/lib/mysql/*

   B. mysql_install_db /var/lib/mysql

4. Comment out the !includedir /etc/mysql/conf.d/ option in my.cnf

---

Kill any existing MySQL Process and then start MySQL using the skip-grant-tables option.

A. Get the exact path of the mysqld daemon:

which mysqld_safe

B. Run MySQL without grant tables:

/mysqld_safe_directory/mysqld_safe --skip-grant-tables &
ex /bin/mysqld_safe

C. Make sure mysql is listening:

netstat -tlpn 

you should see port 3306.

D. If so, log into MySQL:

mysql -u root -h 127.0.0.1

E. Set new password:

mysql> use mysql;
mysql> update user set password=PASSWORD("NEW-ROOT-PASSWORD") where User='root';
mysql> flush privileges;
mysql> quit