Sounds like you are trying to do an adaptation of the "quilt as you go" technique. I quilt on a longarm, so the tops are totally complete when I put them on my frame, so I'm not sure how to do the "QAYG", but you can google it and I'm sure there will be multiple tutorials showing you how to do it. About adding your border...I would do that before I quilted the sections since when you piece a border you do it at a 45 degree angle, not straight across, so the angle of your blocks and setting triangles will be perfect to continue the 45 degree angle. Is there a reason you are doing the quilting in three pieces? I would think it would be easier to put the whole quilt top together before quilting, even if you are squishing it through a domestic machine. Leah Day has great tutorials for quilting on a DSM on her blog.