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I'm not sure you can clean it up - the abrasive particles are really tiny (micron sized) and it doesn't take all that many to keep abrading the steel. I think it's best to just leave the strop as it...

Highly unlikely, especially with a hanging strop - there is more than enough give to the strop to compensate for whatever unevenness a functional razor would have on its spine.
You don't need pasted...