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Question: How to fix a Macbook Pro trackpad where the mouse pointer is randomly moving?
Answer: Today I had a horizontal line not working on the touchpad. I fixed it by removing the touchpad from the laptop (which is very annoying and requires a miniature screwdriver) and cleaned it with a lot of ethanol (ethyl alcohol) and a piece of cotton. I put it back in and it now works perfectly.

I have had a similar problem a couple of years ago and I didn't have a special screwdriver (that I do have now) and I fixed a similar problem with a syringe with ethanol and squirted the ethanol in the crack around the touchpad.

I have seen many people suggest to change the touchpad, but you can try with a cloth and ethanol (remove the battery and charger if possible and wait for 30 min for the ethanol to evaporate).