For normal non card event days, we ask that you specify which common deck you'd like to pick one, should you get a common instead of a rare. You have a 69% chance of getting a common, 30% chance of getting a rare, and 1% chance of getting a limited. You can still include the Rare deck 1 in your request and if you pull a rare you will get a card from that deck, we just need a common option too for if you pull a common.

As for the reroll, that specific reroll type doesn't allow rares to be pulled, only commons. You give back 3 commons and can pick one of the three commons you get, and you can specify the batch or deck you want. We don't have any common for rare rerolls at the moment.
Oh and badges and cards and their teams are different but thats a non issue. Just so you dont get confused