I have not starched doilies but I believe you use a fairly heavy starch mixture to soak them in. Press out the extra between paper towels and have a surface prepared for them. While still wet, they are stretched and pinned to a background until they dry. I would use a flat piece of styofoam or similar to stretch them on. You have to make sure that the pins you use are rust proof! As for fixing the loose thread, I would thread the end onto a needle and weave it back along an edge and secure the end with a drop of fray check?
You could also forget about starching some and applique them to background squares for a quilt.