How to finish the broken edge of my exposed brick wall?
I would consider putting molding around the door and turning over the face of the brick. I would use a jamb-like board perpendicular to the wall and the door, and make its width flush with the face of the brick. Then add a small flat board parallel to the wall overhanging the door side of the jamb by about 1/8 inch and the brick just wide enough to cover the broken edges (maybe 1 1/2 inches). Paint it to match the door and it will look like part of the door frame.

I would fill the gaps between the door and brick with exterior foam sealant. Just be careful since this expands and you don'r want so much that it warps the door frame. Apply it, trim it flush, then do the molding.