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No. Shapesharer does not enter the battlefield when its ability resolves: it was already in play. Stoneforge Mystic's ability only triggers when it enters the battlefield.
Of course, you could use the Shapesharer-Mystic's activated ability after resolving the copy effect.
Note that if you used a different copy card such as Clone, you would get the Mystic's triggered ability, because the copy effect would replace the regular act of entering the battlefield.
No. The new Stoneforge Mystic will not "enter the battlefield", because it is already on the battlefield. In order for "enters the battlefield" triggers to trigger, the object must change zones from some other zone to the battlefield.