Q&A for How to Smooth Glass Edges

Local hardware stores will have one. Emery cloth is just a stronger structured sandpaper with a different chemical compound to do the sanding/smoothing.

It will work but you have to be very careful not to accidentally sand the sides of the pitcher with the sand paper as it will show extremely visible scratches. Practice on something else first a few times to get the hang of it.

Loctite glass glue dries fast and clear. A single drop on a toothpick might run the length of the scratch. Several applications might be required; don't try to do it all at once.

You might consider the technique, of Kintsugi. Applying high quality gold acrylic paint to not hide from the imperfections but (heal) them, visually and energetically.