There is a third, less scientific way to define fall: Phenology, or the study of periodic plant and animal life cycle events. Nature does not check the calendar; it follows the weather.
In this view, fall begins when the environment decides it has arrived.
Under this definition, fall can last longer or shorter than three months depending on the specific climate of a region. In the Arctic, "fall" might only last a few weeks as summer instantly transitions into winter. In tropical regions near the equator, there is essentially no fall at all; the seasons are usually divided between "wet" and "dry" rather than spring, summer, fall, and winter.
No, not technically. While astronomical fall includes the first three weeks of December, most people and all meteorological systems consider December the start of winter. By December 1, the atmosphere generally behaves like winter.
The word "equinox" comes from Latin, meaning "equal night." On the autumnal equinox, the sun crosses the celestial equator, resulting in nearly equal amounts of daylight and darkness (roughly 12 hours each) across the globe. After this point in the Northern Hemisphere, nights become longer than days until the winter solstice.