Or is it like a judge's robe?

I think they're just long shirts b/c Betty and Wilma's dresses have a waistline. I'm going to assume that Barney and Fred's shirts have shorts underneath or something like that! I don't really want to think about it though!

No, Fred and Barney does not wear a dress on every episode

They are definitely animal pelts. This was well understood at the time the show aired.