First off, I'd recommend anyone learn how to do the manual installations of MySQL and PHP. It's not difficult to read how to do it, and it gives you a better understanding of how it works. Having used the installers is probably fine, however I wouldn't try using an older version of MySQL than what is specifically stated as a requirement.
Notice that MySQL said access denied for root with no password. Though it would probably be better to implement the cookie authentication method, in your phpMyadmin config file, you can enter the username of root AND the password. If you don't enter the password, phpMyadmin will try to use no password, which (as you found) won't authenticate and will deny you access.
After you fix this and get logged into phpMyadmin, create a new database for moodle, then click on privileges and create a new user. Don't give this user any permissions when you first create it (i.e. at the root level), but then give the user all permissions to your moodle database except Grant. Then use this new username and password for your moodle installation.
Good Luck