I would definitely line the borders so all will be the same thickness. Also if you are putting sashing in I would line that as well. As to leaving edges without foundation. I would trim to 12" so again, all would be the same, otherwise there might be some of the seams bulkier than others due to catching some of the foundation in the seam and some areas just the string. Leaving some areas without foundation can cause difficulty in quiting, both home machine as well as long arm.
I have taught string quilting and always recommending this way to make the entire top consistent.