use a batting like warm & natural that allows up to 10" between quilting lines- that way you do not have to quilt on the actual aida cloth squares- you can stitch in the ditch around them- use a lightweight tricot or woven interfacing- finish the edges with a small zigzag so they do not fray/pull apart. as long as they are (gently washed) the aida does soften up -- i would quilt in the sashings, areas around the embroidered blocks...i've been doing counted cross stitch since the mid- 80's...and have never had an embroidery floss color run/bleed...i would (guess) there may be some old/off brands that might- but if you used good floss (like DMC) they should be just fine-without worry --- if you are worried about that- after completing (quilted & bound) wash it---with a color catcher tossed in--just in case. i would wash it after completion anyway-that way you remove any soil, oils, dust, chemicals in the fabrics---before gifting it.