I realize that this problem appears to change when you adjust settings in Windows, but the issue is actually a connection/cable issue. There are a few possibilities...
the monitor cable terminator (the plug) is damaged (loose pin, bent pin, missing pin, etc).
the monitor cable itself is damaged (cut shielding/ground, frayed connection at plug, damaged cable within plug, etc.)
The video output port on the video card/integrated motherboard connection is damaged.
Outside interference. An electric motor or fan, power tools, etc. Some device not connected to your computer could be causing this.
The end result is that there is active interference in the signal, and the chosen refresh rate will accentuate how you see it. The reason why you don't see it in the BIOS is because of the refresh rate and display resolution of your system at POST.
You can't try another monitor, so this must mean you can't replace anything. If you can't replace anything, you are most likely stuck with this issue. That particular monitor is more than 7 years old at the time of this posting, it was not an expensive or quality monitor at the time that it was initially made, and attempting to repair it would probably cost more time and money than it would cost to replace it.