What defines a jew

Question

This may sound a little far fetched but I actually know someone who wanted to do this: Can a jewish woman be a surrogate mother ie. have someone else's fertilized egg inserted in her uterus and carry and give birth to this child? If someone does carry a fetus for a non-jewish couple, is the child considered jewish anyway because even though the egg was from a non-jew a jew gave birth to the child? And vice versa- if a non-jew carrried a fetus of a jewish woman and gave birth, would the kid be jewish? Also, while your father is here, can you please ask him why "permanent makeup" was allowed even though it is basically a tattoo?