That is not how it works. They just don't get to put a defense psychologist on the stand. There should be an expert from the prosecution and even an independent assigned by the Judge.
And you wait until the end to bring up she was trying to sell the baby?
And no, you don't need a search warrant at a murder scene.
Also, I can't see the husband not getting the baby immediately. Any Judge would give him custody while it goes through the courts.
And if she was just a surrogate for the couple's baby, the birth certificate is a fraud.