php script automatically creating mysql database?

You cannot create a username and password unless you log in yourself with a user with a higher level. That's usually the root. Creating database is a breeze after using the username and password with sufficient privileges.

<?php

    $dsn = $dsn = "mysql:host=localhost";
    $pdo = new PDO($dsn,"root","");

    //Creation of user "user_name"
    $pdo->query("CREATE USER 'user_name'@'%' IDENTIFIED BY 'pass_word';");
    //Creation of database "new_db"
    $pdo->query("CREATE DATABASE `new_db`;");
    //Adding all privileges on our newly created database
    $pdo->query("GRANT ALL PRIVILEGES on `new_db`.* TO 'user_name'@'%';");

?>

In this script, I assumed your root user is called "root" and its password is empty if that's not the case, change line 4 to match.

you cannot create new user without logging in.

You can create database with root account or user with privileges

When you are connected with the servers root-account (or account with db/user-adding permissions) you can create databases and users.

Adding a user:

<?php
mysql_query("CREATE USER 'USER'@'%' IDENTIFIED BY 'PASSWORD';");
mysql_query("GRANT USAGE ON * . * TO 'USER'@'%' IDENTIFIED BY 'PASSWORD' WITH MAX_QUERIES_PER_HOUR 0 MAX_CONNECTIONS_PER_HOUR 0 MAX_UPDATES_PER_HOUR 0 MAX_USER_CONNECTIONS 0 ;");
?>

Creating a database:

<?php
mysql_query("CREATE DATABASE 'DATABASE';");
?>

Give user permissions on database:

<?php
mysql_query("GRANT ALL PRIVILEGES ON `DATABASE` . * TO 'USER'@'%';");
?>

You definitely can! You can use PHP.

First you need to connect to the database. Then you need to run a SQL query on that database that creates the database. (which is essentially what cpanel does)

EDIT: Updated since my_sql is deprecated

Heres a stackoverflow post on how to do it...

Can I create a database using PDO in PHP