If this were my quilt, I'd definitely use the same color thread on back and front--it's so frustrating to have a bit of the other color showing through to the other side if the tension is off even a hair! And as a wall hanging, it would rarely be seen and anyway would look fine.
I'd outline each of the inner designs with thread matching the white/cream background and staying 1/16-1/8" from the edge, filling in the light area with a tight stipple/meander. I'd break thread for each one--it's only 12-14 thread pairs to bury later (14 if the bib and carriage white inner areas are done separately). A tight wavy line in the small white border would set off the squares. I'd try a dark grey or even deep green in the maroon border so the design would be visible but not intrusive; maybe straight lines, or meander, or even free form flowers. (See blog The Inbox Jaunt for some ideas for easy free motion designs.) And I'd try out all these ideas on scraps first, to make sure my mental vision translates well to fabric.!
Have fun sewing, and know whatever you do will look great in the end, because if you hate a portion you can frog it and change to something else! And you've done such a good job so far, you obviously have good design sense.