I just went through this process a few weeks ago. FIrst, go to the inspection station in SW as soon as they open or make an appointment for early in the morning. You don’t need any paperwork to get your car inspected. If your car passes, they put an inspection sticker on your car and give you the paperwork. Then drive around the corner to the DMV. Tell the person at the front desk you need to convert your out of state license to DC, and register your car. They can do both these transactions at the same window. I was done with everything in under two hours, it wasn’t that bad.
As for the documents you need, I HIGHLY recommend using the DMV documentation guide for BOTH registeration and getting your new ID: http://dmv.washingtondc.gov/info/DocumentVerificationGuide.shtm
For your new licence you need proof of ability to drive (your MD license) PLUS proof of SSN/ID which cannot also be your MD license. So you need to bring a US passport, birth certificate, and/or SS card. For proof of residence, all I needed was my lease. I also didn’t have any utility bills, but I bought a cable bill just in case (I didn’t need it, my lease was enough). It sounds confusing but as long as you follow the Document guide you should be able to do all transactions in one day. Also I totally did not do all these things within 30 days of moving here (more like 6 months after moving here, and no one cared) Good luck!