Yes. You cannot do commercial photography in the gardens (bridals, etc.) without permission and I believe there's a fee involved. There is no photography permitted on the second floor of the museum.
Definitely would. Lots of events going on there. Plus you will need three permission.
I would if only as a courtesy and there are special events going on often. Wednesday is free admission day although the entrance fee on other days is modest.
I would. I don't know what their policy is, but many places have specific rules regarding commercial photography
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