Putting "Washington" for the city should be fine. If you look at the actual form DS-82 for passport renewal there's just one field called 4. Place of Birth (City & State if in the U.S., or City & Country as it is presently known.)

Bonus if it is ever relevant: we're also in Washington County. Washington City and Washington County and the District of Columbia used to be three separate things!

Where (approximately) in NW? NW is pretty big so a rough location (like a nearby street intersection) might help you to get somebody who's willing to help.

It seems that Virginia has recently authorized same-day voter registration (as in, this is the first election that has it): https://www.elections.virginia.gov/registration/same-day-voter-registration/

You will need to look around on that website or use their "Contact Us" phone number to call and ask if there is a minimum residency rule there.

No problem. You'll still want to send in your voter registration application online at your convenience so you will be able to vote in our next election.

Welcome to DC!

https://www.dcboe.org/Voters/Register-To-Vote/Register-to-Vote lists a wide variety of documents that are acceptable for proving your residency for voter registration

The hitch for OP is that you must live in DC for 30 days prior to an election to be eligible to vote in it (according to that same page)

Please note that you might not be qualified to vote in this election. According to https://www.dcboe.org/Voters/Register-To-Vote/Register-to-Vote :

> You may not vote in an election in the District of Columbia unless you have been a resident for at least 30 days immediately prior to the election

You can try calling the BoE to double check but that may be the limiting factor here.