I've seen quilts done with cross stitch blocks, and it is totally do-able, and seeing your blocks, will look amazing.
I completely agree with suern3 about adding a stabilizer- the lightest weight woven fusible interfacing you can find. Not only will it keep the blocks from distorting, but it will also anchor all of the thread ends on the back.
Because you will most likely have large areas that are not quilted where the cross stitch is, double check your batting distance requirements. You might even want to overstitch some of the outlines in the cross stitch through all of the layers to hold every thing together and add dimension to the designs.
As for whether you quilt before or after assembling, that would depend on your preference. Since you enjoy handwork so much, I would lean toward a quilt as you go method (again, suern3 is right, lots of youtube videos out there).
As for how washable it would be, it depends on how colorfast your threads are. I would test them for bleeding before tossing them in the wash. Other than that, it depends on your level of bravery. Personally, I've done cross stitch designs on clothing in the past, and not had any issues. Once it is all quilted and stabilized, if the threads don't bleed, I would not hesitate to wash on a delicate cycle with cold water, then letting it air dry. I would not use a dryer on anything with fusible interfacing, since heat can cause it to separate.