"Do not ask what they think, ask what they do!"
Translation:Non chiedete cosa pensano, chiedete cosa fanno!
Yes, the tu conjugation in negative imperative always uses the infinitive: see http://italian.about.com/library/weekly/aa011900b.htm
Yes; you can read other posts above, or https://www.thoughtco.com/the-imperative-mood-in-italian-4072739. The imperative mood is a bit of a patchwork, and the negative one for "tu" forms with the infinitive.