In order to add a waypoint to an existing path, you must select it before placing another point, otherwise FRED's going to create a whole new path. Same thing with the waypoint editor: you need to have a waypoint selected in order to be able to edit its name.
To order ships to follow waypoint, you give them waypoints or waypoints-once orders either as initial orders (ships editor) or via SEXP (event editor).
EDIT - Since 3.6.16, you can also move waypoints around during the mission itself by using the set-object-position SEXP. It's very useful if you want your capship to have semi-dynamic behavior, but it's a bit tricky to do, so I'd recommend getting more used to FRED before doing that.