Can Shapesharer copy an enters the battlefield ability, like Stoneforge Mystic's?
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.