You're question was answered but I would add that it is definitely a better option to park on post where it's free and vehicle is safe from break ins etc. Does take like 40 minutes to get to King St. Station to grab the metro into DC but worth the extra few minutes to avoid the hassle
The REX comes on post, with multiple stops. The hospital, commissary, etc..
Yes they do. The Belvoir Buses run daily M-F to Springfield so that you can reconnect with the train routes going through the Capital region.
I don't live in that area anymore I'm sure they have public transportation though
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