I would highly recommend using a lightweight fusible interfacing on the knits. Pellon makes several weights. It is a bit challenging when sewing the cotton to the knit then pressing the seam flat but it can be done. Be careful with the blends, sometimes the high heat from your iron will "glaze" or even start to melt them. Using a pressing cloth, such as a piece of muslin will help to protect them.
I think your friends daughter will love and cherish this quilt no matter what you do so don't fret too much about the stripes not lining up or being straight. I think that adds visual interest. Spend some time browsing Kaffe Fassets patchwork gallery, he does such fun stuff with stripes.
http://www.kaffefassett.com/Patchwork.html