This popped up in my search terms today:
“does vargo hoat kill jamie”
I can say, very confidently: no, he does not. I’m confident about this because Jaime was very much alive last we saw him in A Dance With Dragons, whereas Vargo Hoat died at Harrenhal during A Feast for Crows. It was a messy, lingering, painful, humiliating death. The Mountain and his soldiers had their fun with him. But he’s gone now. Jaime saw what was left of his body.