As far as the hair and bag, I don't always know what bag my child is bringing with her because she changes them several times daily. I would have to make an effort everyday to notice it. I don't. I may be a horrible mother but mornings are hectic and I don't notice everything my child has with her.
The texting thing, I have not idea what really happened there.
As to when someone saw her last or what had happened in the arguement, DT was not there and those details can change with the people who actually witnessed it.
If one of my children were missing, I would pick up my son at the park as well. I'd want to make sure I knew where they all were.