Short answers, pulled from the story.
Weather refers to the state of the Earth's atmosphere at a specific place and time, described in terms of temperature, humidity, cloud cover, and stability. Climate is the averaging of atmospheric conditions over longer periods of time. Weather covers day-to-day temperature, precipitation, and other conditions.
Weather is driven by air pressure, temperature, and moisture differences from one place to another. These differences arise from the Sun's angle, which varies by latitude, and the strong temperature contrast between polar and tropical air. Earth's tilted axis causes sunlight to strike at different angles through the year, producing the seasons.
The atmosphere is a chaotic system, so small changes in one part can accumulate and magnify into large effects across the whole. This instability, combined with measurement error and incomplete understanding, means it is theoretically impossible to make useful day-to-day predictions more than about two weeks ahead.
In 1281 the Kamikaze winds destroyed Kublai Khan's Mongol fleet and saved Japan from invasion, and in 1565 a hurricane destroyed the French fleet, ending French claims to Florida. Hurricane Katrina scattered over one million people from the central Gulf coast, the largest diaspora in United States history. The Little Ice Age caused crop failures and famines, including a Finnish famine in 1696-1697 that killed about one-third of the population.
The coldest air temperature ever recorded was -89.2 C at Vostok Station, Antarctica, on the 21st of July 1983. The windiest recorded place is Commonwealth Bay in Antarctica, where gales reach 199 mph, and the greatest twelve-month snowfall, 31102 mm, fell at Mount Rainier, Washington.
The most successful attempts at influencing weather involve cloud seeding, used to disperse fog at airports, increase winter mountain precipitation, and suppress hail, though the evidence for efficacy is inconclusive. In 2008 China fired 1,104 rain dispersal rockets from 21 sites in Beijing to keep rain from the Olympic opening ceremony on the 8th of August.
Yes, weather follows many of the same physical principles on other planets but at different scales and chemical compositions. Jupiter's Great Red Spot is an anticyclonic storm known to have existed for at least 300 years, Titan has rain of liquid methane, and Neptune has measured wind gusts up to 600 metres per second.