Yes please stabilize your fabrics...especially the stretchy ones...Use NON WOVEN stabilizer...I think Pellon makes it, but you don't want the fabrics to stretch while sewing so the non-woven ones hold it in place.
I made a t-shirt quilt and had no problems cutting or sewing with the stabilizer. Put in on before you cut a block.
I think you can use most of the fabrics...just know that they will all wear differently when washed. But I wouldn't worry too much about that. Aging quilts are so comfy and loved.