There are more examples on the page. There are people knowing these behaviours, who try and use the common impression to their advantage. For example the people who are dishonest, who have trained themselves, to maintain eye contact, and so inspire unfounded confidence. This tactic can be defeated sometimes by remembering the principle, if it sounds too good to be true, it probably is.
I thought it was interesting that eyes up right was visualizing an image, while up left was remembering a sound. Sounds reasonable the way the brain is divided right/left, but I wonder if it is the same for left handers. Or somebody that is ambidextrous

I never could control my eye-rolling which often got me in trouble at work. And there were so many reasons to roll them.

I don't think it changes depending on which hand is your dominant. You look to the left when your trying to come up with a lie becuase the left side of the brain handles creativity. Where as the right side of the brain is more analytical, deals more with facts and math.

It does not seem to matter wether someone is right sided or left sided or even sided. You get the same responses. The differences to this would show up if the person had a brain abnormality or brain injury.

psy and behaviorist make it sound so simple and transparent- Pity that it is not (or not)

I often don't make eye contact, but it's not because I'm lying, it's because I'm usually nervous, also I'm sometimes go cockeyed. I had a friend once who was permanently crosseyed. That doesn't seem like a full proof method in the least.