Yes, the bottom floor is open to the public and the top floor is open as long as you're escorted by a member of the staff.
Yep! Make sure you check for tour times though - it's first-come, first-serve, so make sure you get there a little early. Due to the size of the house, the tour size is limited and people are sent in as others come out.
You are encouraged to go in and learn.
Yes - check the website for tour times
They have a great introductory movie about Hamilton, and then you proceed upstairs.
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