Interesting. I'm actually learning to use the other hand precisely because I'm also learning to play the piano -- Chopin's preludes have very demanding left-hand parts, so I figured I would probably benefit from strengthening my left-hand muscles in any way possible. I also plan on purchasing a left-handed classical guitar for this reason as well.
I don't think what I'm doing is a standard practice, I've heard of certain stress and 'baoding' balls being used to strengthen muscles (video: http://www.baodingballs.com/baoding_ball_techniques.html ), but I think that's about it. Normally, pianists just practise playing études and doing run-ups with scales.
That said, I have noticed a difference in my fretting hand as a result of trying to become ambidextrous, so it hasn't all been for naught. :) As for what I do with my left hand: anything that I can do with it -- I brush my teeth with my left hand, hold my cup of tea with my left hand, and in easier, non-technical classes, I copy down notes with my left hand. I have been doing this for about 2 years now.
