@ Ikemitsu - What type, size/capacity is the main storage device (HDD, SSD) in the Mac Mini?

Is the storage media known to be reliable?

Does the Mac Mini boot OK from the installation media (CD or USB stick)? If it does it might be worth doing a 'health-check' of the storage media via SFSDoctor to ensure that it is not the cause of these issues.

I cannot remember having a flashing 'globe' on the screen if the system ever wouldn't boot, ... yes I have seen a flashing apple/mac monitor flash onto the screen, but I don't know what the flashing globe is you refer to, others might, but it's something out of the ordinary to me!

Usually I have a Dual-Boot system for a Mac Mini, and Tiger is also the best choice IMHO.

You can download Mac OSX Tiger from Mac OSX Tiger 10.4 Retail DVD download (that's the direct weblink) and update it as required to 10.4.11 once it's re-installed - if you don't already have the Mac OSX Tiger installation media.