Hello all!
I know it seems like the touchpad is possessed. I can assure you it is not. I know that it is a highly frustrating problem. Please trust me on this one and just try this setting change. SmartSense is the culprit. I've seen the following changes correct this issue first-hand. Please give it a try! 
Click Start > Type "mouse" > Click Mouse
Click the ClickPad Settings Tab
Click the ClickPad Settings Button
1 Click Sensitivity > 2 Click the Gears symbol OR Double click Sensitivity
If the cursor moves when you don't want it to, or moves too much, Move the notch closer to Heavy Touch
If you are experiencing undesired clicks, 1 Click SmartSense > 2 Click the Gears symbol
If you are experiencing undesired clicks, move the SmartSense setting notch closer to MAX.
This is one of the most common adjustments. I've seen it fix the problem everytime.
Remember to Click Apply to Save the Changes. If you change a setting and find out you don't like it but can't remember what you changed, just click the Defaults button (bottom left) to set all of the settings back to default.
I hope this helps! Like I said, I've seen it fix this "randomly clicks" problem several times. The setting defaults are designed to suit the majority of people but may not suit EVERY one out there. It's a very easy fix though.
Please let me know if you have any questions