Not comfortable walking distance.
Yes... and Uber or take public transportation(subway) to the Met. Lots of hotels in center city.
"Decent" and "Close" are subjective. There are hotels within a 5 mile radius. I would search hotels within the radius you consider "close" and filter based on the amenities you are looking for. The Met is in a very convenient location in many ways.
I think the hotel at Broad and Fairmont is open if not then the one near Broad and Chestnut
No
Not really
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