Just wanted to add how much pain you are in with kidney stones depends on the size of the stone and other factors. My worst pain (decribed above) was when I moved a large stone from my kidney down towards my bladder where it got stuck in the ureter. I had to have surgery to remove the stone--but it passed miraculously with no pain or even notice from me before the surgery.
So, it's variable. I would say it's the pain up near your kidneys that is the worst and most like labor...I think actually passing the stone out of you body is not like birthing a baby's body at all (though I am sure it's still plenty painful for some). Good luck with both of those things.