I want to bring residents from a nursing home. Is there an area where they can rest and eat a snack brought from home?

Of course, on the 6th level, there's a restaurant but also a sitting area outside. All accessible by wheelchair.

There are expensive food options on both the 6th and 1st floors. I would eat before you go in and drink water from the plentiful fountains

Yes, there is place to sit and relax and no, you can't bring in any food. In fact, security at the door is pretty serious. But you can always talk to customer service via and explain you situation. They are very understanding and helpful.

As far as I know they only allow food bought in the museum.

They could sit at one of the many tables in the cafe on the 2nd floor right above the main entrance. Food and drink cannot be brought in.

There is some bench seating around the grand staircase, and an outdoor patio on the 6th floor. but I'm not sure about the policy of outside food. There is a restaurant named MANNA also on the 6 th floor

Yes, many places for rest and an elevator to get from floor to floor