Short answers, pulled from the story.
A gear train is a machine element formed by mounting two or more gears on a frame so that their teeth engage. The gear ratio, determined by the ratio of pitch circle radii or tooth counts, governs the speed and torque relationship between the input and output gears.
The transmission of rotation through toothed wheels can be traced to the Antikythera mechanism of ancient Greece and the south-pointing chariot of China. The Renaissance scientist Georgius Agricola illustrated gear trains with cylindrical teeth, and the later development of the involute tooth yielded a standard gear design providing a constant speed ratio.
An idler gear is an intermediate gear that transmits power between an input and output gear without contributing to the overall gear ratio. Its primary functions are to reverse the direction of rotation of the output gear and to bridge the physical gap between distant shafts without changing the speed ratio.
A hunting gear set has tooth counts that are relatively prime, meaning no common factors, so every tooth on one gear contacts every tooth on the other before any pairing repeats. A non-hunting gear set lacks this property, causing certain tooth pairs to contact more frequently, which leads to uneven wear.
The 2004 Chevrolet Corvette C5 Z06 six-speed manual transmission has ratios of 2.97:1 in first, 2.07:1 in second, 1.43:1 in third, 1.00:1 in fourth, 0.84:1 in fifth, and 0.56:1 in sixth gear. The car also has a differential with a final drive ratio of 3.42:1.
The crankshaft-to-camshaft gear ratio in a four-stroke engine is always 2:1, meaning for every two revolutions of the crankshaft the camshaft rotates once. This ratio is required by the four-stroke cycle and applies whether the engine uses a timing belt, timing chain, or meshed gears.