Can't tell you about the gun laws but I can tell you that most of the shootings in DC occur in the SE quadrant. Stay out of that part of DC.
NE is not much better.
SW is extremely small (Arlington County,Virginia used to be most of SW) but some parts outside the federal areas are a little rough.
The real dividing line once you move away from the mall is Rock Creek Park.
To the west is benign, to the east is not so nice.
No guns or ammunition (not even a single round) in DC.
They LOVE to gig people and charge them, so do NOT give them an opportunity.
Virginia is actually more convenient than many parts of DC to stay in, and you gun is perfectly legal here.
You could lock it up in the room safe or front desk safe at your hotel.