I was just wondering if we are supposed to sew close to the edge of the quilt before trimming it and applying the binding. I have noticed several videos on you tube about applying binding but no one ever mentions if you are supposed to sew down the edge of the quilt

I do. I find that it prevents slipping and tucks.

i do because i quilt on a frame and bast the edges as i go (most of the time) it does make applying the binding easier having everything nice and flat on that edge

I learned to do this watching a Fons & Porter quilting show once. It is especially helpful when I attach the binding to the back first and roll it over to front.

If I have quilted on the LA I have tacking stitches all around the edge anyway. When FMQ on my DSM I usually have quilted so close to the edges that it isn't necessary. Hand quilting, it all depends how densely I have quilted the border and how "loose" things appear around the edges. I will make a judgement call if I find it necessary or not.

A lot depends on the tightness of the weave of the fabric and if it has borders or not. I am currently handquilting a quilt whose top was made of loosely woven feed sacks. A lady at the Senior center donated it to benefit the center. It had no borders. I took it home and stay stitched he whole outside before washing it to be sure it would not fall apart. I then pressed it and added a solid border on all four sides. I currently have it more than three quarter of the way handquilted and it is doing fine.
Staystitching is a good idea if there are no solid borders or if the blocks have a lot of bias edges. It is mostly a judgement call.

I usually quilt to the edge, so I generally do not stitch around the outside again.

If the quilting is dense, there will be less chance for slipping, I don't. If the quilting is looser with bigger areas not quilted right at the edge, I do.

I do. I always use a narrow zigzag stitch. It flattens the edge and makes it easier to apply the binding.

I attach the binding, then trim