Mouse moves by itself down to dock (even when physically unplugged and no network connection) and starts clicking open the applications in a row! Always does the same thing every few hours. What is happening?

Question

Craziest thing ever.
I cannot find this happening to anyone anywhere online, but this is seriously the weirdest thing I've seen in 20 years.  I've done everything I can think of to try and prevent this, EXCEPT for two things... one of which is being tested right now:  login as different user and reinstall the OS.  For the past 30 minutes, I've left it logged in as another user to see if it happens again.
Things I've tried:
- New mouse
- New keyboard
- No mouse or keyboard
- Repair permissions
- Reset PRAM
- Turn off any extraneous applications that launch at login
- Firewall blocks / Stealth mode
- Remote management OFF
- No network connection at all
No matter what we tried above, it still happens.  Again, I am currently logged in as a completely different user to see if it happens here.
If you were to see it in action, it would freak you out.  The mouse literally just starts jumping around on the dock opening applications.  We have not found a pattern for when it actually happens. It just happens.
Any ideas, friends of the forums?  Thank you in advance.

Answer 1

Thank you so much BDAQUA and RRFS. 
I can't confirm it, but I think I just figured out what is causing the issue.  This is a school computer, and there is a SMART Board hooked up to the Mac via USB.  Thanks to BDAqua's post, I realized the "ghost" vanished and never came back when the SMART Tools application was closed.  Through all of this troubleshooting, I failed to realize a GIANT mouse was still attached to the computer.
I have not returned to the classroom to confirm any of this, but I would bet that either something is on the SMART Board causing the "touch" to occur or the software just needs to be reinstalled.  Again, I've confirmed nothing, but it sounds like it MUST be related to the touch screen SMART Board.
Thank you all who gave this any thought. I apologize for not physically removing all peripherals... I hope this works.  I'll be back there tomorrow.  If you don't hear from me again, that was definitely the issue.
Thanks again!