One small correction on the documentation you need – you need BOTH a U.S. Passport and your SS card. The first is to prove your identity, the second is to prove your SSN. The link talula provided is very helpful in walking you through what you need (just take the < off the end of the address if you follow it from this page and it will work).
I also second that you need to have your insurance set up in DC and proof of said insurance before you can register your car here. Good luck! I’ve heard mixed results from people about their experiences, but if you have all the documentation you should be okay.