In His Hands or on His Hands?

No, "on his/their hands" is idiomatic. "In his/their hands" is not. 2 means "He has a competitor to deal with".

Do you say in your hands or on your hands?

I am holding something 'in my hand' is what we say. If something is 'on my hand', it is on the surface of your hand. For instance, a bird might land on your hand, or you might have dirt on your hand.

What does in your hands mean?

COMMON If something is in your hands, it is in your possession or under your control. This move will leave too much power in the hands of the party leadership. You have to trust your surgeon.