I hope you figured this out, but since I don't see any other replies I thought I'd shave what little I know.
All your problems come down to corner recognition, as you seem to recognize, and issues with corner recognition almost always are due to marks in the margins (between the corner markers). You can see this clearly in your Corners.png image, where the signature has crossed into that region. The Coners4.png seems to show the same thing happening because of the poor quality of the scan/printing producing extra black marks, and that may be true for the others too (it is impossible to tell from the images since the offending corners aren't included in the image). In all these cases you can see the green lines, which should show the corners, are not finding those corners correctly and distorting or cutting off parts of the scan.
I don't know if the border/box you have around the form is contributing to that... I have used the circle markers mostly and those would be affected unless the border was clear of the whole band occupied by the corner marker.
Beyond that, to make this work for you you will need to find some way to get rid of these kinds of marks. If the extra marks/smudges around the edges of the image are a common source of the problem then a better scanner or printer that doesn't put marks in the edges might be necessary. Or you could find some way to automatically 'clean up' the images... by having software white over certain regions of the jpeg for example. Shrinking the form portion to avoid things like the spilling over signature might also be necessary.
I am just another user, so I may be missing some other solution that Formscanner has, but this at least is the source of both the issues with marks and with the bar code, so you hopefully at least have some idea what needs fixing.
All the best!