You need to starch before cutting the fabric for best results. If you starch heavily first, your cuts will be accurate and the fabric will not stretch while you are handling it.
There are some patterns for which the templates do have a right side and a wrong side. I don't think yours is one of them, though. Is the not-fitting problem just with size being a little off rather than the shape being completely wrong? If so, it is a problem of accuracy rather than right side/wrong side of the template.
Edit: On taking a closer look at your pictures, I do think that some of those templates must have a right side and wrong side. Try turning the fabric pieces over (wrong side up) to see if they fit that way. If they do, it means the template has a right side and wrong side. In those cases, it's important to mark the right side of the template (the side that will be facing you when you cut) before cutting.