That is, if the hot wire touches a metal part of the fan which is not supposed to be energized (say the metal housing), and the fan is not properly bonded, then there will be no ground-fault path back to the source to cause the breaker to trip. This could mean that the housing of your fan would be energized with (in a home in the US) 120V, and would shock anyone who touches it.
First, NEVER work anything hot. There is no such thing as a safe voltage to work energized. Even a simple home receptacle circuit can easily kill you (and it happens with alarming regularity).
That said, ALWAYS work something AS IF it were hot (for the exact reason you said, accidents happen). This means connecting the safety measures first. The safest order to connect wires is: ground, neutral, hot. The safest order to disconnect wires is the opposite: hot, neutral, ground.
Half right, half wrong. It would not short if it touched the metal box, and would not trip the breaker. That part is right.
However if it touched YOU not having the ground wire connected actually helps you. It means the electricity does not have as a good of a path back to the box, and you'll get a weaker shock.
With the ground wire you'll get a much higher shock, but it won't trip the breaker!! Humans have too high resistance to trip breakers, you'll just keep getting shocked.
So it's kind of a wash - with the ground, it's more likely to just short against the box and trip.
Best is just to always wire things as if they are live. Only touch the wire if you have to, always use insulated tools, even if the power is off.
If that guy was connected to the earth he would get fried. Note that you still have to equalize whatever potential there is between you and the system, like they do in that video. Same reason why you should ground yourself and the hardware before working on your PC. Ever had a tiny spark hit you when you touched something (neither you or that "thing" being connected to the electrical grid, just static electricity e.g. from walking on the carpet)? But without connection to ground the only thing that happens is equalization of the potential between you and the target but no further flow. If you are connected to ground on one side and to a source that keeps creating an electrical potential, like the electrical grid with a power plant somewhere, there will be a continuing flow. If you are not connected to ground there won't be.
