Q. Is it permitted to read the M’gillah anywhere else other than in the synagogue?
The synagogue is the usual venue because it enables us to make the miracle known (pirsumei nissa). The presence of a minyan guarantees an audience for this purpose, though the M’gillah may be validly read without a minyan.
The reading should normally be done by an adult (Shulchan Aruch, Orach Chayyim 689:2), though there are authorities who permit a child under Bar-Mitzvah age to read the M’gillah if there is no adult able to do it.