I cant' believe your Grandma won't listen to you and would let your DD CIO!! That's so disrespectful to not only your DD but to you and your DH also.
YOUR baby, YOUR rules.
I probably wouldn't leave my DD there if I knew she might do that.
As for screaming/crying in the car for 20mins... I know what you mean... it's awful... my DD did it for 1hr once when she was 6wks i was on my own it was heartbreaking even stopping the car didnt help! but yeah making a baby CIO in their cot on their own is VERY different!!!