Along with the 5770 I have a Kona 3 which works with all 3 NLE's.
I use FCP7 on this machine by far the most and have no issues with the 5770 there.
Avid 6.01 via the Kona is basically very smooth in terms of simple clips on the timeline.
I use Avid profesionally and it feels as quick as the Avid Mojo or Nitris hardware.
However I have had a few spinning beachballs when messing about with the Colour Corrector and other FX. Its not perfect but then the MacPro 1.1 does not run a 64bit kernal and Avid is not supported on a 1.1 so perhaps not suprising.
I am new to Premiere but so far I have found it runs fine until you start using the Kona for output. Then it goes clunky and stutters and is not great.
To be fair I have not really used Avid or Premiere in anything other than testing and playing around with simple sequences and fx so nothing conclusive to draw from my comments other than generalisations.
How or if any of my clunkiness is related to the 5770 who can say, - however it runs very quietly, does not seem to get hot, and is pretty much the only upgrade worth considering as far as I can tell as nothing else that works offers any improvement or doesnt run FCPX or has other limitations.
In terms of running other apps outside of NLE's it seems not to miss a beat - I cant think of any issues. I run 2 24" Samsung monitors from the mini DVI ports and all seems fine.
The monitoring on FCPX is not great but drivers are still beta - I never use it I cant get on with the whole thing, in broadcast where I work it has been totally ignored - I dont know a single edit house or facility that has even considered it.