Is it permissible to shake hands with a non-Muslim, an innovator, or a transgressor, and to be gentle with them?

The default ruling is the permissibility of shaking hands with everyone, unless there is a reason that necessitates prohibiting touching them, such as a man touching a non-mahram (definition) woman. The situation varies depending on what serves the greater good. If shunning them leads to their guidance, then there is no harm in it. However, if associating with them and winning their affection is more conducive to their guidance, then one should act in a way that is more likely to achieve that.