I mean, I'm not entirely sure that burning down Ford's Theater would have much of an impact here - if the Confederates captured DC they are doing so early in the war, or toward's the middle. There's still more than enough time for Booth to try to assassinate Lincoln in another location; assuming he lives and is still radicalized (which he likely would be). I mean, it's not as if his attempts are intrinsically tied to Ford's Theater, after all.
Now, the rest of this queston is actually a bit more interesting. The loss of DC would be seen as a huge shock to the Union, and would definitely give the pro-Confederates in Maryland an opening; whether 'officially' suceeded or not, the state is probably going to end up largely in Confederate hands until they're driven out.
I would imagine that the Union government reorganizes in quick order, likely in Philly (a city that is far enough away from the front that it's not going to be threatened unless things go REALLY South, but also close enough to the front to be able to manage it). The entire debacle is going to be a huge black-eye to the Union and Lincoln, but likely not a fatal one; especially if it happens early on.
Now, you're missing the better analogy here, though. Instead of a reverse Sherman's March, the Confederates sacking DC, burning it and a White House, will likely be compared in American mythology to the British doing the same during the War of 1812. Assuming the damage is near total, by the time the US pushes the Confederates out, they're going to have a decision to make. Rebuilding the city is going to take a significant aount of time and money, and it will be years before it's ready to house the government again. During this time, the Union goverment will be happily taking root in another city. Coupled with the fact that the location of DC itself was a compromise between the North and the South, you're going to see a significant amount of people arguing that it should just remain in whatever new city it ended up in, rather than rebuilding DC. Meanwhile others will say that American honor demands the rebuilding of DC, bigger and better than ever before, to to prove that the US can't be pushed around. I give it a good coin-flip over which of these options get taken; though my preference would be Philly (or Chicago? There are a number of options) becoming the permenant seat of government.
And yes, the Constitution states the capitol needs to be a Federal District; but it doesn't state how big said Federal Distict is - so it's entirely plausible that a new, smaller, Federal District is carved out in a different city and DC reverts to Maryland (once Maryland, fo course, is fully subdued and reconstructed).
On a side note - you may want to look into the masterful Until Every Drop of Blood is Paid, a timeline on this board, where the capture of DC (and a few failed attempts to reclaim it by the Union) play a major role in helping to radicalize the war further.
