This is your chid. Her welfare is paramount. You should converse as much as is needed for you to feel safe in leaving your daughter with them.
What were their reasons for not wanting to care for their granddaughter in their own home? When I took my first granddaughter for three days a week, ( my son’s child), we had to make changes in the house. We were fortunate to have a ‘ playroom’, which our children had, but still needed gates etc. To be honest, it’s was so much easier for me to care for her in my house.
So, to answer your questions:
1. Of course you can ask them to look after her at their house. It’s your call.
2. Yes, you should have that final chat to make sure you all know what you’re doing.
If you’re not happy with anything, don’t leave her. You’re not obliged to do anything at all when it comes to your child. Being ‘ fair’, although a nice thing to do, isn’t necessarily best for your child.
Parents-in-Law. What do/did you call them?