Cursor moves when mouse is plugged in

All these seems unreal. Yesss, ghosts are vibrating your mouse =)

Maybe anything vibrates near the mouse. If not, and if nothing helped,
then simply try to boot to another OS.
It will help you to understand if the problem is in your OS or not.
If it continues to vibrate, then the problem is in your house.
Otherwise (if the problem is in your OS), run ProcMon.exe to watch what's happening when it vibrates, and also check autoruns (using Autoruns.exe, not msconfig), and check driver/program startup order with ProcExp.exe .You can download them in Sysinternals' website.

If after all this your mouse continues to vibrate... then ghosts are vibrating your mouse :)
Good Luck!

Clean.

First, make sure to clean your mouse optical hole thoroughly, ideally with lens cleanser and lint-free tissue. Different lighting conditions (day/night/fluorescent) between your work and home could cause different behaviors. Modern mouses are very sensitive and it's not always obvious it's a dusty/dirty problem until you clean it.

Known issues

What is your mouse model? Maybe there are known problems or device incompatibility with it.

Disable.

Next, I would try to disable EVERY "Mice and other pointing device" in Windows Device Manager. Maybe some unused device driver, or the touchpad itself could be interfering with your mouse.

Isolate the problem

Approach the issue scientifically. It's okay at work, bad at home.... surely, something is different between those 2 places! Try both battery or wall power. At home, try another room away from equipement. Try outside. In the park. Try half-way between your work and your home. Try at work in a meeting room. If it vibrates everywhere except your own work desk, ask your boss what he did to your mouse.

New: Corded vs wireless

Is this a wireless mouse anyway? Try a corded USB mouse. (Or vice-versa).

The problem you are facing is really challenging. Your hardware is sensible to some interference with some signal that is present in your house. If you have a laptop, you could try and disconnect all the electrical stuff in your house, or even turn the electrical power off.

The signal could also come from outside, say a mobile communications antenna. It would also be worth trying with a cordless mouse. Maybe the mouse cable acts as an antenna.

I would add that since other laptops don't have the problem, your laptop's EM shielding is flawed. I would report that to the producer and ask for assistence.