Dean
I have recorded a short video of how I would go about this - maybe I am missing something here, if so let me know and I will think again, however I did the two things here and they seemed to work for me on a mock up of what you are showing in your PDFs - if you could send me a VCL of the Bridge Abutments then I can take a look at your data specifically to see where the differences are
1) You can create a Rectangle at Elevation 605
2) Once created make it a SUrface Model called Datum 605
3) Use the Surface Intersect command to calculate the point on each line of your Bridge Abutment at the elevation 605 - you select the end of the line as the coordinate, the Datum 605 surface as your target, the bearing would be the bearing of the line of the abutment and the vertical angle is the angle between the start and end points of the line (you can use Point To Point to calculate the Vertical Angle. Execute - this gives the point on line 1 - Repeat for each line to compute the 4 points and then draw a linestring between the 4 points to compute your shape and check it.
a second method I used was to create a Surface from the two lines and then to creayte a contour at Elev 605 (this you did also), and it creates more than one line segment between the two lines, because it crosses a Triangle Face - maybe when you created your Datum Surface you had it creating many points along each side ie many triangles and that is why you had many vertices - I just had the one which is easy to delect if you want to get rid of it, or you could just draw a linestring for the rectangle between where the contour hits the edges of your abutment. If you need to change how your surface is created for the Datum Surface - change the Project Settings, Computations, Surface to stop it from densifying the lines creating nodes in the TIN model.
Hope that this answers the question - let me know if this helped
Alan