Had Lee won at Gettysburg he would have been in position to attack DC. That threat alone may have prompted the Union to negotiate a peace treaty. A defeat in DC almost certainly would have resulted in a final peace, or at least a years long peace between wars.
Almost certanly not. Had Lee won at Gettysburg he would still have been in the middle of Union territory. He would have used up much of his ammunition with no line of supply back South. He would have had tens of thousands of wounded to care for. And D.C. was ringed with forts and had a garrision as large as, if not larger than his remaining army. All Lee could do, win or lose, was go home.
Interesting because DC was one of the most fortified places on earth. Fortifications and earthworks protected the city.