Magnetic poles always exist in pair.
Answers
If a bar magnet is broken into two or more smaller pieces the smallest piece is magnet since it Will have both north and south poles to attract each other
Because magnetism is the result of electric current interacting with magnetic materials. It takes a minimum of two charged particles moving to produce a circuit.
Origin of Permanent Magnetism
ALL magnetic fields are the result of electric current. From the magnetic fields at the galactic scale down to the magnetic fields of your refrigerator magnet at the atomic scale.
No matter how small we divide a magnet, we can not produce a magnetic monopole. A magnetic monopole would be the magnetic equivalent of an electric charge. So they are considered not to exist. Since it takes two charges moving to create a circuit, theoretical magnetic monopoles are not needed to explain magnetic fields. Just that one accept every scientific experiment ever performed and come to the correct conclusion that all magnetic fields are produced by electric circuits.