If you want to check if a wire is live/hot, in a domestic circuit (max 240 Volts AC), a mains test screwdriver might be enough. They are usually insulated to 500V, and light up when the tip touches a live wire, while your hand touches the contact on the end of the handle. It detects potential to ground, which is all it needs to give you a shock if you touch the wire.
Be sure you set the dial to read the voltage “high”, rather than “near”, or you could burn up your meter before you can blink. In other words, if you expect to measure 240V, set the meter to read on the 1000V scale, not 200V. Most meters have a AC and a DC scale. Be sure to use the correct one.
The difference between the neutral and the ground is that the ground is grounded in your house. The neutral is grounded somewhere, but there can be a substantial and variable voltage between the neutral and ground. When I was in graduate school, some boneheaded professor, for reasons that were never apparent to anyone presumably including himself, tied the neutrals in his lab to the building ground. It took several of us with sensitive equipment a few days to figure out that the building ground had been floated, i. e., was no longer earth ground. So for some time there were about half a dozen pipes pounded into the ground outside various labs, and we had to run copper braids from our labs out to our own “personal” grounds and electrically isolate our equipment from the building.
I’ll take a crack at it. The idea of grounding audio equipment is to shield it from ambient noise, such as the buzz of the house current and random pops and hums. To do that, you want all of the grounds connected -together- by the shortest rout possible. In high-end speaker wire, you have 2 conductors inside, surrounded by a braided conductor called a “shield”. The shield is the only “grounded” lead. Same with mic wires, etc.
High end wire tends to be a heavier gauge because it has lower resistance. Connecting grounds together by the shortest route also lowers resistance. If you have a wire from point A to point B that has a certain length, and the wire is a perfect conductor (for the sake of illustration, but it does not really exist), then there will be no resistance. If you ground end A and then place a noise source in the middle, then no noise will make it to end B. In the real world wires do have resistance, and so under the same circumstances, there will be noise at and B. How much noise depends on the resistance of the wire.
After you connect all of the grounds together, then you want to earth ground them. You may do that is several spots to minimize resistance to ground and equalize the “ground potential” throughout the system.