[No, this will be a new tunnel at ground level. The existing tunnel will be abandoned. Found a 9 page PDF WMATA document portion describing
King Street Station access improvements. The focus of the project is on improving access to and through the DC Metro station, not on so much on the Amtrak/VRE station. The new tunnel is called the VRE Passageway Tunnel which shows the primary motivation for constructing the tunnel. What is interesting is that the tunnel will use a knock-out panel in the wall at the King Street Metro station which opened in 1983, so this tunnel has been in the planning documents for a
long time.
What the tunnel and elevators will do is not only make it easier for Amtrak passengers dragging luggage with them to get to the King Street Metro station, but also easier to walk to the hotels and businesses across the street and to Old Town Alexandria.
I do not believe that there are plans for a platform on the 3rd track. The long term plans for the Long Bridge to AF Interlocking segment, AFAIK, are for 4 tracks which would be separated into 2 passenger tracks and 2 freight tracks for normal operation. We will see what the plans are for the 2 track Long Bridge bottleneck when the current study group posts documents for public meetings and comments, but that is likely a ways off. (and getting OT for this thread).