One would think so. Some of it would certainly be draining back into Lake Ponchartrain, but the bulk of it seeminly would find its way to the Mississippi. I haven't heard anything about what's happening to water levels in either of those lately.
It depends. The bulk of it will not. In Central Mississippi and most of Alabama, the water will drain to the Gulf through the Pearl, the Tombigbee, etc. The Mississippi drainage generally runs between along a line between I - 55 and the Mississippi River - the area of Mississippi getting the least rain. Jackson, Hattisburg, Meridian all do NOT drain to the Mississippi.
Farther north in Alabama, and for all of Tennessee to be hit, the water will drain into the Mississipp via the Tennessee and Cumberland Rivers via the Ohio, so it will be a long time before that water gets back down to the lower Mississippi. Most of the Georgia rain will flow south through that state via the Chattahoochie and Flint. Only the extreme NW corner drains into the Tennessee River and thus the Mississippi. A small portion, given the footprint of the storm, will drain into the Atlantic rather than the Gulf.
In Louisiana itself, much of the drainage will be into Lake Ponchitrain or the Pearl. Just west of Baton Rouge, for example, the drainage is into the Amite River, not the Missisippi.