3 6970's comfortably outperform 2 580's and depending on the game will ever outperform 3-way 580 SLI. Here is just one of many benchmarks that have been done between different setup variations of the two cards: http://www.anandtech.com/show/4254/triplegpu-performance-multigpu-scaling-part1.
The question is would 4x6950's beat 3x6970... answer is I don't know, and I haven't seen any benches on it either. Though performance scaling does diminish as you add more GPU's and tri-fire seems to be more stable than quad-fire.  not even taking into account a suitable mobo.
For three displays the 580's loose at again as having 25% less VRAM is something you have to consider when running at those resolutions.
All out for a 3 screen display at the $1000 price mark I would recommend tri-fire 6970's from what you mentioned above but would suggest you also consider tri-fire 2GB 6950's as the performance is not too far off the 6970 and is significantly cheaper.