MySQL on AIX operating system I could see any proper documentation which answer these questions   

1.Is MySQL database and cluster supported on AIX operating system?

2.Is MySQL database and cluster free for commercial use in any private enterprise application ?
Referring to https://www.mysql.com/support/supportedplatforms/database.html the answer is no, AIX is not a supported operating system for MySQL database however you can install it through yum, for information on how to install yum on AIX refer to https://www.ibm.com/developerworks/aix/library/aix-toolbox/alpha.html

Concerning support you should address this with MySQL Technical Support (ref. https://www.mysql.com/support/) as they are the ones who would ultimately need to decide if they will provide support for your installation, which they may refuse on the basis that you are not operating their product on an officially supported / certified operating system. Concerning pricing, no, MySQL for commercial purposes is not free (ref. https://www.mysql.com/products/).