Certainly looks like some sort of hardware issue. The first thing I would do is disconnect the internal hard drive since a failing hard drive can sometimes prevent booting to external media. That quickly removes one road block.
You can try running the Apple Hardware Test which you can boot from the 2nd DVD which originally shipped with the Mac from the factory to see if any hardware issues are detected (hold down the 'C' key with the DVD inserted). If you have the Late-2009 Mini and if it had macOS 10.12.4+ installed at some point, then it may have the online Apple Diagnostics available (as well as Internet Recovery Mode --- Command + Option + R if the internal HD is connected -- to access the online High Sierra installer).
I personally like to boot a Knoppix Linux USB stick in order to better & more easily check the hardware of these older Macs. macOS tends to get into trouble when it encounters something unexpected whereas Linux may be able to finish booting which can give you a better idea of the situation. The Linux boot log can sometimes be very informative as well unlike the Apple system logs. Plus Knoppix has a utility installed by default that can check the health of the hard drive if it is connected.
The missing row of pixels for each line is interesting and may suggest a memory issue or GPU issue.
Edit: If you cannot get the Apple Hardware Test or online Apple Diagnostics to work, then you can try creating a bootable Apple Diagnostic USB stick by downloading the proper diagnostic image file from the following site (all links are to actual DMG files hosted on Apple's own servers, so there is nothing to worry about):
https://github.com/upekkha/AppleHardwareTest
The DMG files should contain a PDF file with instructions for creating the bootable diagnostic USB stick which I believe utilize the "Restore" functionality in Disk Utility.
