A question...what does Taylor mean by "careless mans careful daughter???"

I don't think she had family problems in real life, but in the song she did, and like you said she saw her parents fighting when she was a little girl in the music video. So what she means by "a careless man's careful daughter" is that her dad was careless and fought with her mom, and didn't really care what she saw or heard or anything, so (in the song) she was always careful because of the way she was brought up around all the negativity and she didn't want to get in a bad relationship like her parents had. So when she finds her guy in the song he "learns her secrets and figures out why she's guarded" (because of her parents fighting and all the stuff I mentioned before) and says that they'll never make her parent's mistakes. So they'll actually make it and be happy. And although many of Taylor's songs are based on her life, this one isn't.

she may or may not have had family problems we dont really know. But knowing taylor and her soulful music she may have witnessed it and saw that and wrote a song for awareness on this kind of stuff and how kids or adults or teens go through this kind of living

Yes because in the video you can see that there is her parents arguing!!!!!

taylor means her dad is careless and she is careful