I own a 20-inch mid 2007 iMac (iMac7,1). I was able to boot ubuntu-14.04.2-desktop-amd64.iso from a DVD and USB flash drive. I used instructions posted here to create the flash drive. The DVD verified OK, but the flash drive listed an error in 1 file. I did successfully install Ubuntu to an external disk using the DVD. Have yet to test using the flash drive.

If you have Lion installed, did you make a bootable USB flash drive using the OS X Recovery Disk Assistant v1.0? If not, then does your mac still have its hidden recovery system partition? You can determine this by holding down the option key at startup. If you do, try to see if you can boot to it. Making a USB bootable recover disk assistant will allow you to reload OS X Lion from Apple's web site.

Without a working optical drive, you could still install Windows. This would require installing a free copy of VirtualBox on OS X. This option is "off topic" and it would be better to ask at the site Ask Different. Actually, the instructions can be found at Boot camp install of Windows 7 issue, no bootable devices.

To install Ubuntu using a DVD, you would have to repair your optical drive. A factory refurbished Apple SuperDrive 8x DVD Burner Drive can be purchased for $8.75. This would be an upgrade to your existing optical drive.

If interested, a service manual can be found at this site. Here is a link to the actual manual: Mac mini (Mid 2007) Service Manual.