What happens with the "Homing Pidove" attack if I choose not to shuffle my deck?
Homing Pidove tells you to look at the top card of your deck. You don't get to keep that card, so you put it back onto the top of your deck. Then you can choose whether to shuffle your deck or leave it the way it is.
Source: Roaring Skies FAQ (2015-05-07), TPCi Rules Team (2015-05-07)