Your favorite member of the Bat-Family, and why.
Wow this is actually one of the hardest questions; at least I'm getting out of the way first right? Wow, okay... Well my first impulse is to say Bruce Wayne; main character, most Bat-stories center around him and he is the reason I am drawn to Batman stories in the first place.As a child I had this deep abiding sense of justice and an awareness that the world and the systems of the world weren't inherently just. And Batman understood that. He understood that sometimes you had to step in and set things right because the people that were supposed to do so wouldn't or couldn't. Even though I disagreed with his no kill policy I respected it because it showed that Batman wasn't out to destroy or undermine the system as much as he wanted to support it, rehabilitate it. Get it on the right track.
But I think I'm going to go with Cassandra Cain: I have loved every single one of the additions to the Bat family, Dick grayons, Jason Todd, Tim Drake, Stephanie Brown, and Damien Wayne (who I must admit I wasn't too sure about at first). But Cassandra Cain is the 2nd girl on the team and the first person of color and her story is so poignant. Plus her ability to read people purely through body language is something that I've always wanted; it's almost like telepathy :)
What series best showcases the family?
I'm going to focus on the movies/television series as that is the way I generally consume my Bat-media. I would say the series that best showcases the family was The New Batman Adventures, simply because it has the most characters in it and that was satisfying for me. I always wanted to see more of Dick as Nightwing and Tim as Robin as well as outlying characters like Renee Montoya, Lucius Fox, and Kara Zor El.
Your new addition to the Bat-Family (i.e. Supergirl, Swamp Thing)
I think my favorite additions to the Bat-Family have already been made and they are Terry and Matt McGinnis and Damian Wayne. Technically the first canon biological children of Bruce Wayne (even if it is in the future) there is soooooo much story potential that was never mined except in fan fiction and we all know how hit-or-miss that can be. Then Damian came along and at first I was really skeptical especially with the then-recent "death" of Bruce, but he got under my skin and I am more than pissed that he's been killed off so quickly. There's a lot to be explored about Bruce Wayne's psyche when introducing biological children I don't think it can be overstated that for such a loner, Batman has managed to gather around himself an effing huge family and my favorite stories are the ones that explore the Bat-Family dynamics. I even have a community called Bat Family Chronicles on FFN, so if you come across a good fic PLEASE message me on G+ so I can add it. Good writing is hard to find even in the comics much less fanfic.
If the Bat-Family were to fight it out like in Marvel Civil War, what would the sides be?
It would really depend on the situation. Off the top of my head I'm pretty sure that Cassandra would go where ever Bruce goes and Damien might be torn between Bruce and Dick if they are on opposite sides Jason is his own man in every way so he would probably end up creating some third faction that no one is happy with and if Dick and Barbara aren't together then Oracle might end up on Jason's team.