Yes. If you plan on walking the exteriors a stroller should be fine. The interior of the mansion does have a narrow windy stairway that wouldn't allow for a stroller though so keep that in mind when going.
Should be no problem, but like others have said there is a nutty staircase in the mansion tour.
Yes. Well worth trekking the baby along. Make sure you see the slave cabins and the trainers cabin in the property. Also a beautiful public garden nearby. Can't recall its name but they would know in the gift shop if you are interested in that sort of thing.
Yes. If you do the mansion tour, be aware that you would have to go up a long, and kind of narrow staircase. There are sidewalks around the grounds and an elevator in the building with the gift shop and restaurant.
Yes
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