First, the reason you can not change starting points is usually because of the cutting order. It is setup so that both parts are cut with 1 pierce point. You can sometimes get different pierce locations if you line up the 2 objects you want, select them both and click on the common line cut icon.
As for changing the pierces, do you mean objects with multiple pierces? If so right click on them in the Parts list and click on advanced edit. This will allow you to change the cut order for parts. Just remember you want your part connected to the material it is cut from until all other internal cuts are made.