So I have a new theroy about Kate. I think that when Emily got Catherine out of the cocoon, the morph, or whatever was left of it, retreated to the next best thing, her womb. It formed a new cocoon there, or used it as a cocoon. From there, it started to unify with the fetus. This also had a shadowing effect on Catherine, which most likely left her braindead, but the morph wasn't able to assume control over her body anymore, just keep her body alive.
I don't know why it took so long to unify with Kate, or why she took so long to develop. After she was born, the morph was no longer keeping Catherine's body alive, so she died.
Kate is a unique existence among shadows. Rather than mimicking a human like a morph normally would, she developed like a human baby would have. She is not quite like the other shadow children, nor is she like the adults. I think she will continue to grow and develop over time and will at some point be more powerful than normal adult shadows.