It really depends on the temperature. I guess by "power" you refer to the ability of the magnet to attract (for example) an iron nail. The more accurate term is magnetization.

How long a magnet takes to lose its magnetization is a function of what the magnetic materials are made up of, and how high a temperature you heat it too. At high temperatures, any magnet will lose its magnetization rather quickly (in a matter of minutes, if the temperature is high).

Magnets stay magnetized until something destabilizes them. The constant jostling of molecules (i.e. heat) slowly disorganizes the magnetic domains in a magnet, and this happens faster the hotter the magnet.

Ultimately, a magnet will align itself to any pre-existing magnetic field, such as that of the earth. I don't know how long it would take for a magnet on your refrigerator to demagnetize - probably longer than a human lifetime.

This is a really great question. It would make sense that a magnet would operate like a battery and would run out of power. Actually, it turns out that magnets don't operate like batteries. A magnet works by the atoms lining up in a piece of iron or steel. When the atoms are aligned, north and south poles are created, resulting in magnetism. The way in which an atom looses power is when the atoms come out of alignment. This can happen when a magnet is dropped. Therefore, a magnet will not ever loose its power unless it is dropped, or experiences some other force which misaligns the atoms. I hope this helps.