If the vacuum cleaner is powered by electricity this is used to turn the electric motor which is needed to create the suction or vacuum. That motor either has permanent magnets inside it, or the current passes through many "turns" of wire to create magnets when you switch the cleaner on. Early experiment in electricity showed that if a wire is moved in a magnetic field some current is created. Similarly, if a magnet is moved near to a wire the same result occurs. So, yes the cleaner does need a magnet.
In the motor used to turn the fan that creates the suction, though you might find magnets as closures too.
air presure pushes dirt into the vacuum cleaner.
to clean faster
Yes, the phrase "his mouth was a vacuum cleaner" is a metaphor. It is used to convey that someone is eating or speaking voraciously or loudly, likening their mouth to a vacuum cleaner.
No At least, not unless it happens to have a very large magnet in it for some reason.
A magnet or electromagnet.
Some kind of magnet or magnetic field is necessary for a motor to operate. Alternating current vacuum cleaners (that plug into the wall) have motors with electromagnets. Direct current vacuum cleaners (that plug into the car's cigarette lighter socket) have motors with permanent magnets.
That depends on the distance to the vacuum cleaner. The closer, the louder.
Given the history of the vacuum cleaner, the Hoover is the best choice for that trade or common name.
Vacuum cleaner
Although it is called a vacuum cleaner, it does not actually create a vacuum. Inside the cleaner there is an electric motor which drives a fan blade. This fan blows air out of the vacuum cleaner and this causes air to be drawn in to the cleaner at the other end. Thus the cleaner works by creating an air flow, not a vacuum.