Does anyone do all basic arithmetic in their head? No, so the answer to your first question is that they certainly did some basic arithmetic on paper.
As far as a scale of skill in mental math... I would sincerely doubt this exists. This would have to be similar to a World Records maintaining group, and maintain a record for all "famous" researchers. Of course, then they'd have to define famous.
I would be willing to bet that you will only find individual stories of the mental math capabilities of mathematicians or physicists, but with no organization maintaining a ranking system or scale.
Related: Euler carried a math problem to 50 decimal places in his head: http://library.thinkquest.org/22494/stories/Euler.htm Then again, he was Euler. :)