That poor, slow, fat little owl...
UPDATE: OP has a Pikipek and a Yungoos, and says he will upload pics if they evolve. Will update with stats when they are available.
The Trainer Card is now referred to as the Trainer Passport, possibly to help facilitate island to island travel, but that has yet to be confirmed.
Pokemon Refresh is available at any time once it is unlocked, and not just directly after battle as some news sources erroneously stated. It is accessed via the main menu, and is located right next to the Trainer Passport icon.