No coat check
No
I didn't see one there.
I didnt notice one either, but I dont think so because the place where you get entry tickets is a really small side structure. Also I when you get inside the mansion, theres really only seating furniture to interact with. I agree with Corinne, your car is the best place!
Not that I'm aware of. You could leave your coat in the car when you are inside then grab them for the garden. I'd still wear a sweater inside since it's such a cold day.
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