I would complete my quilting first, just as I would for any other project.
I would construct my ruffle within a casing, as I showed in this thread:
Give me your Ruffle Making tips please!
People more talented than I can carefully pin everything and go around it just once with a straight or zig zag. I have to open up one side, sew that, fold it back over, and then hand finish on the back just like I do my normal binding.
The easiest way to turn corners is use something like a quarter as a circular guide. You can miter but it is an additional thought process to make sure fabric doesn't get tucked in/out the wrong way. Really, that slight curve is highly recommended and it is rarely noticeable, even with very square/grided designs.
Depending on what you use for your ruffle (you might want to treat lace differently, a carefully done miter isn't so bad, especially if you gathered your ruffle yourself. Stabilize the gather and then take it out where your seam will be, whether diagonal or straight. Sew closed, regather, and finish with invisible edge.