Poorly. I mean, nobody's buying that the show is called "Supergirl" for any reason other than that this is a pre-existing character. As the producers said: Shes Supergirl because shes Supergirl." What happened in the first episode is that Kara had a perfectly reasonable complaint and a fictional character shut it down by saying that she didn't have a problem with it and she's the boss so that's that. I think it's totally understandable that someone who has no prior attachment to or even knowledge of the character would object to suddenly seeing a female version of Superman being called "Supergirl".
Yeah pretty much. It it fine when Kara is portrayed as a teen, but I am always weirded out by a grown ass woman being called girl.