I learned a few things in the making and quilting of my red and white quilt last year. I've never really bothered with trimming threads from the back of the quilt but as you can see from the photos below untrimmed threads can stand out.
The threads showing through is only really a factor if you are using a light fabric. The other niggle I noticed was that there was some shadowing of the red through the white fabric (I don't have a photo to illustrate this) this was caused by not trimming the red fabric back adequately when I paper pieced some of the blocks.
With this quilt I decided to attach the hanging sleeve using this tutorial which was quite successful although the line from the blind hem machine stitching was visible. In the large photo you can clearly see the line across the top of the quilt.
I wasn't going to wash the quilt as I didn't want to risk any of the reds running (even though I had washed most of them previously). The other thing I noticed with this quilt was that in some places you could see the shadow of the backing fabric (a red and white toile). In the scheme of things none of this really matters because I think it is still a very a nice quilt and I'm happy with how it turned out.
If I were to make another quilt like this then I would consider using two layers of batting as this would make the quilting stand out really well and there would be no shadowing from the backing fabric.