No. The fact that her great-grandparents were 1st cousins would not have caused her death.
Grandparents not great grandparents. Much closer.
if you have half expression of a particular genetic defect that runs in your family you have a great chance of giving full expression to a child if you are conceiving with a relative.
There is an interesting story about how it looks like Meghan Markle’s ancestor was a product of incest between two half siblings.
mental health issues may run in her family