Monitor turns off for approx 1 second when fridge starts?? Why??

If three different cables result in, good, bad, and ugly, then the problem is probably not power line problems.
I am not getting a DVI to Cat5 converter.

How are the Cat5/6 cables running - in parallel with the circuit wires for the fridge, or not? What I am thinking here is that the startup of the fridge is somehow inducing a spike via the electrical cord or circuit to the fridge, and this is feeding back to the Cat5/6 cable to your video, somehow upsetting things. They say when wiring a network to cross power circuits at right-angles with to the Cat5/6, and to not parallel a Cat5/6 cable near a mains circuit, or in the same conduit (part of this is common electrical code - shorts and other electrical issues, mixing low/high voltage in same cable path a no-no, etc - but part is the interference issue). Perhaps the "surge" is altering (power-factor?) the waveform on the mains, and this inductance in the Cat5/6 is upsetting the frequency of the video signal, and dropping the sync-lock at the monitor?

The fridge is in a different area of the house. The two cat 6 cables are running parallel also along side a usb cable.

I have today tried to shorten the Cat 6 cables from 10 meters to 5 meters. This doesn't seem to have fixed the issue.

When we design DVI products a difference in wire length is very critical. The wire pairs from the connector to the IC are very critical. I don't know why I keep all the PCB traces the same length to within 1mm and then the signals get divided between two cat-5 cables that may be the same length.