I would read it as #3: Permanently reduce the Hardness each time an object is struck (to a minimum of 0)
It does not restrict the weapon used; instead it merely says "Reduce the hardness...by 1" without specifying a specific weapon or attack. Everywhere else that I can see that has a duration (e.g. Bless) specifies it somewhere, and things that have a permanent duration (e.g. Weapon Focus feat) doesn't have a duration and is presumed to me infinite.
As for whether or not it stacks, I would interpret this as a penalty to the object's hardness score:
Penalties are numerical values that are subtracted from a check or statistical score. Penalties do not have a type and most penalties stack with one another.
and since the wording is "whenever the object is struck by the brawler's unarmed strike" without any restriction on how many times this can occur, I would conclude that the penalty would be able to stack since there is no restriction on that.