I have used Acetone in the past to remove old glues, as it is a strong solvent. You must use it outside, as it is not good to inhale the fumes... I will generally use a stiff toothbrush and try to lightly scrub it off.
As far as the repair goes, I would use a couple of very tiny drops of superglue (I use the gel and not the runny fluid type) The whole edge does not have to be glued to hold it in place...
Craig