Thank you for the information.
It could be that your daughter just reacts differently to events and that is why she doesn't show much fear or anxiety. Some people react with shock or even laughter when they see something traumatic or are part of a serious situation, such as a funeral. And shock can present as being detached and unresponsive. So she may be completely normal in how she seems after a trauma.
The problem with playing to a crowd is a possible issue. But at her age, it could be that this is how she learned to get what she wants. She is young and at that age, both environment and personality play a part in how children act with others. That is not to say this is something she is picking up at home. It could be that she sees it in school or has been exposed in some other way. To test it, you can try to redirect her behavior and see how she reacts. Talking to her directly about it may help (in a gentle and non confrontive way). Give her examples of how to handle her feelings and her need to please and see if she responds.
Self centerness is very normal for an only child. They are used to the attention being on them all the time and there is no competition. This may be easily addressed just by socializing her more than you normally do.
Sociopathic behavior is very rare and it is unlikely your daughter inherited it from her father. Much of sociopathic behavior is developed during childhood during severe neglect
and other abuse. There are hereditary factors, but there is no guarantee that because your child has a father who is sociopathic that she will be as well. There are many children born in families with sociopathic parents and they turn out very normal. More likely, your daughter is either reacting to her father's issues or she is exhibiting some symptoms of either adjustment problems or possibly this is just her personality. It may help to work on her issues through behavioral changes (with a therapist helps) just as you would any other behavior issue and see what occurs. At this point, her behaviors can be addressed easily and effectively.
I hope this has helped you,