Where can I get the image of the installation disc that came installed (mountain lion)? If I open the App Store, I can see the OS X Mountain Lion with a price and not "installed" or something like that. I don't want to pay for the OS that is already installed by default. Isn't there a way to download the image, an utility to create it or something like that?
There was no DVD for Mountain Lion. Since Lion arrived with the App Store as a digital release, the core OS is not something you re-download from the store. You don't need to pay to re-download, but you do have to go through the Recovery HD to initiate the download.

When you buy an upgrade from Snow Leopard to Lion or an upgrade to Mountain Lion - that installer can be re-downloaded, but when the OS is included with a CPU purchase, you don't need to have an Apple ID or account to re-download the OS. You just enter recovery mode and run the installer. It downloads the package after verifying eligibility by checking the serial number on the Mac hardware.

If you interrupt the process, you can grab the installer from the drive - but it's not something that the system was designed to deliver to you as a downloadable package. It was designed to download and re-install on an "as-needed" basis without needing an account, a password - just an internet connection and some time to wait for the download.