I haven't thought this through properly, but to increase the size of the transformed area, could the coordinates of the control rectangle be massaged a bit, e.g. scaling it up to the edge of the image, before passing it to the AForge algorithm? This would provide some attempt at transforming the surroundings. I don't know if naively scaling it would change the perspective effect though, I haven't looked at the maths.
A cheap fix for the anti-aliasing would be to enlarge the picture x2, perform the transform, then shrink it x2 again. There is a property on the transform that "specifies if bilinear interpolation should be used or not" which defaults to "true" so maybe it is doing it already.