Does the Museum have a "Metro" Stop of its own?

No it does not

Good afternoon, Your best option for Metro is the Huntington Metro station. That is the metro yellow line. Once at the station you would take the Fairfax Connector Bus #171. It will take you to the museum and it runs on the half hour. The same bus line will also return you to the Huntington Station. This bus line runs seven days a week. Please refer to: fairfaxconnector.com or call 703-339-7200 for current bus information.

No, the DC "metro" does not go directly to the Museum. The closest stop is Franconia Springfield at the end of the Blue Line. From there you can get a bus to the museum.

Yes

Yes it have a stop