I saw Carley's battery problems as the writer's idea to get us to "like" Carley and imagine her as a potential love interest for Lee to set us up for the fall in Ep 3. That said, I've on occasion had to flip batteries for my wife. Is it a stereotype or a reflection of society forcing gender roles (men must possess mechanical knowledge and are rewarded by women for possessing such)?
I didn't see enough evidence in my Carley playthrough that Lily thought anything of Lee other than a potential ally/leader. She spent most of the time shouting at him. It's possible, though.
There isn't much in the game that leads us to believe that Carley has any interest in Lee until she's about to get shot in the dome. I mean, I was pretty sure that's where they were going with her, but they were timid because of the race issue. In my playthrough at least, there's no real evidence to show either character had any feelings for her until that moment in Ep 3. If they really meant for us to feel her loss they bungled it.
Ultimately, Lily is a canon character and they had to get her to Woodbury on time. All else is in service of such.