Some might say that vertical and horizontal pulling aren't complimentary, but I think they are. Anyway you make your lats stronger will help you pull in either plane.
So that answers your first question as well. Any type of pull-up or chin-up is good. Since you got off the machine and are doing freeweight, any grip is superior to what you were doing.
"It's okay to have idols as a kid, but when you grow up you have to become what you dream about." -Education of a Powerlifter
"Though one were to defeat thousands upon thousands of men in battle, if another were to overcome just one - himself, he is the supreme victor." Dhammpada verse 103
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