If the cotton lace were of any width I would pre shrink it. To preshrink it I would do it in the sink, plunge it into very warm if not hot water with a good dose of vinegar to set the color. If you have ever had a trim shrink noticeably or have the color run, you would never risk it again. I pre wash all of my cotton fabrics. I made a First Communion dress for my goddaughter that shrank 2 inches when I was finished. I was sick. A dress with over $150 worth of fabric and laces shrinking will do that to you.
Fusible interfacing, I would pre shrink. I am not personally a huge fan of pellon, it does not stay fused in my experience. To pre shrink it you can soak it in the sink and hang to dry. That is how I pre shrink the fusible interfacing I use fusible woven interfacing. Usually a product known as German Interfacing and I am getting ready to order some interfacing from
http://sewexciting.blogspot.com/ to try.