In the early 1800’s a series of earthquakes reversed the flow of the river for a short time. How long did this flow like that for? What would happen if the river permanently flowed like this? If that is at all possible.
To get you off to a start, those Earthquakes occurred in 1811 and 1812 at the New Madrid (Missouri) Fault. Here’s a Wikipedia article about it and it has a mention of the Mississppi River reversing.
http://en.wikipedia.org/wiki/New_Madrid_Seismic_Zone
Well, at least you now have a definite date and location of those quakes.
I don’t see how or why the Gulf of Mexico would start draining into Minnesota…
If the Mississippi flowed backwards permanently, Minnesota would change from “the land of 10,000 lakes” to “the land of one big lake.” 
Seriously, the fall in land elevation is too great for the Mississippi to flow north on a grand scale. If an earthquake raised a section of the river bed by, let us say*, ten feet, that would force the river to flow backwards for a while. The rise in the ground would have the same effect as building a dam across the river. The continuing inflow of water would eventually rise by 10 feet and the flow would resume in the normal direction (south) again. An alternate possibility would be that the water pooling behind the dammed area would find another route before the 10’ high earthquake created dam was breached. This would be the process by which the course of the river is changed.
Minneapolis/St. Paul is generally around and elevation of 950 (give or take a few dozen feet). For the Mississippi to flow backwards on a permanent basis, some place along the course would need to rise above that elevation. Even in this case, there are too many areas near the river that could act as the overflow point before Minnesota flooded and became a new Great Lake.
The Okavango Riverin southern Africa that flows into the Kalahari desert and all of it either seeps into the ground or evaporates. It can happen, it just won’t with the Mississippi in the foreseeable future.
- A 10 foot vertical shift in ground elevation would not be impossible, just considered an extremely major event.