If you were new to physics, and someone told you that average velocity was defined to be $\Delta x / \Delta t$, you might ask them what this expression had to do with the usual meaning of the word "average", since on it's own, it's not obvious. In the last integral in the question before the last equals sign, you can see that you're adding up an infinite number of velocity values, each having a weighting of $dt$, and dividing that sum by a sum of the weights ($t$). It retains the usual meaning of "average", so both are acceptable as definitions. But the integral itself is equal to $\Delta x$

The Franck-Hertz experiment is done with DC current, not AC current, but I'll allow that it could just switch polarity at 60Hz (USA). However, if this is truly the same effect as the Franck-Hertz experiment, then each gap should represent 10.437 volts of potential difference for singly ionized mercury vapor; however, we should also expect that the center of the bulb would be noticeably dimmer than the ends.

@MartinUeding The gas necessarily has to be ionized to some degree; otherwise it could not carry a current through it. Since a plasma is rather chaotic, collisions between ions and electrons can cause both additional ionizations as well as only exciting the outermost electrons to many of the allow transitions without ionization.

The people in the chain do not have to turn their individual directions. The chain is a pattern made up of the collective positions of the people. Since some people enter the grass before others, the pattern created by the chain of people alters it's tilt angle, but the directions of the individual people in the chain do not change. The chain appears to be moving in a direction that's different than each person individually. Which means each person's neighbors are either ahead of them or behind them a little.

@Landau are you asking why these are the correct formulas? If so, then the answer has to do with how you change basis states in a Hilbert space from a position basis to a momentum basis or the other way around.

Fluorescent lights work exactly because the gas inside is ionized into a plasma. Hg vapor is ionized, which produces UV photons when the electrons re-combine with the + ions, and these UV photons strike the phosphor coating on the inside of the glass, and the atoms in this phosphor coating undergo fluorescence, absorbing UV photons and re-emitting visible light photons.