Your Mac mini is a 2007 model that maxed out at OS X Lion 10.7.5. It does not support Mavericks (10.9.x) or even Mountain Lion (10.8.x).
OS X installer disks are based on the same OS they install. That is, when you boot from a Mavericks install disk, you're booting a stripped down version of Mavericks. That's why your Mac mini isn't able to boot from that install disk.