What you want to do is difficult, but not impossible.
First, start with the zoning process. Look up what DC Zone you are in (R-1, R-2, R-3, etc). You need to determine several zoning related issues: setback requirements, height limits, maximum coverage area, and number of units allowed. Your Zone will let you know all these things. Should you want to exceed any of the stated limits, you will have to get a Special Exception from the Board of Zoning Adjustment. The BZA is a long process, but they are generally helpful as long as your request has some precedent.
Based on your description, I think the most challenging aspect of the pre-construction process will be extending the back of your house. This will be an issue for maximum coverage area and setbacks. You will also want to check out the "pop-up" zoning. There is talk about limiting the ability to construct pop-ups, so you will likely stand a better chance of approval if you start soon.
Digging down in the basement will involve an engineer and the permitting process. This is typically just a question of cost versus return. (Digging out the basement is very common in the DC area.)
I'd suggest doing the research yourself to determine if your goals are feasible. Feel free to send me an email with any other questions. [email protected].
