How does current flow through the earth and back to the supply to break the fuse?

Originally the "Earth" connection was just that, a metal spike driven into the ground. The earth itself conducts a little, especially when moist, and the electricity was left to find its way back to a similar spike at the supply.

The power station would supply three phases, but its neutral point would be connected to a large earth-spike system. The consumer Neutral would typically be taken off at the point the three-phase supply was split off into single-phase for local distribution, and at that point also connected to an earth spike.

This did not always work well, especially in sandy deserts or on solid rocky ground, where the ground was very dry for a long way down. It was also prone to energy spikes on the Earth wire during thunderstorms, when the ground nearby was struck by lightning. The separate earth and neutral circuits could also create large loops, where regular geophysical earth currents set up voltage differences which both affected the supply voltage and induced large DC currents in the neutral line, reducing their effective AC capacity.

Nowadays the three-phase line is often given a fourth Neutral wire, although the earthing system may remain in place. In the UK, the consumer earth typically connects back to the plumbing rather than an earth spike as such, on the assumption that the copper pipework is well grounded. If plastic pipes are used, it's back to the old ground spike (to which any runs of copper pipes still need to be connected). I am not sure about other countries - some, such as the EU (and the UK for lighting circuits), try to avoid earths altogether and rely on electrical isolation of metal boxes to avoid short circuits.

So yes, where there is an Earth it should always connect back to the Neutral.

There are two ways the metal housings of equipment are grounded for safety depending upon the power system which can vary by country.

In the US (TN system) the neutral conductors and equipment grounding (safety) conductors are connected together at the service panel to a bus which is I turn connected to Earth by a grounding electrode. So if the live wire touches the metal housing the fault current flows back to the panel grounding bus blowing the fuse or tripping the breaker without having to actually flow through the earth.

In some other countries (TT system) the metal housings are connected more directly to the Earth so that the fault current flows through the earth to back to the supply. In the US this is not permitted by the NEC out of concern that the impedance of the Earth can reduce the magnitude of the fault current so that the fuse won’t blow or breaker trip. Countries that use this system get around the problem by using residual current devices (RCD’s). These are similar to GFCI’s and trip at much lower currents than standard fuses and breakers.

Hope this helps