Virginia does not recognize a permit from Alabama.
Link to relevant Va. State Police web page
Open carry is permissible in Virginia. Const., Art. 1, Sect. 13. There's no "resident/nonresident" distinction as to open carry. You are also permitted to have a firearm secured within a container inside the car without running afoul of the concealed weapon statute. Va. Code sect. 18.2-308. "Secured" does not mean "locked".
My understanding is that possession of firearms and pretty much anything related to firearms is prohibited in DC, notwithstanding the Heller decision. Slim strips of DC are located South of the Potomoc, along the George Washington Parkway, which appear to be, but are not, in Arlington County. So I'd stay off of that stretch that runs between Memorial Bridge and Alexandria if I were you.
Take a look at this thread and see if it gives you any ideas:http://forum.opencarry.org/forums/showthread.php?93581