Can you applique and quilt at the same time?
You can use the quilt as you go method and just have your applique pieces glued, or pinned in place. Then when you have all of your layers together you can sew around the outside edges and applique them down— and this will also be quilting it for you.
Do you square up a quilt before adding borders?
Once your piece is quilted, you need to make the edges even and square before adding your binding. It is easier and more accurate to trim the quilt sandwich if the three layers are sewn together at the edge. Otherwise, the bottom layer can shift and you end up with edges that are not the same.