YSK that if your turn signal is blinking unusually fast (like 3 or more times a second), it generally means you have a bulb out.
Why YSK: Your signals alert other drivers as to what you're doing; a signal bulb costs a few bucks and is usually a quick and easy repair to do yourself (consult YouTube); and any place that regulates motor vehicles probably requires you to have working turn signals. So knowing when and how to replace a burned out signal bulb can save you an interaction with law enforcement.
Adding: You can diagnose which bulb is out by turning on your hazard lights and checking all four corners of your car. It'll be the one not flashing.
This is also probably a good time to check your brake lights. Put something heavy on the pedal or have a friend hold it down and check that all three brake lights illuminate. Replacing a burned out brake light is also usually pretty cheap, quick, and easy.
I don't believe so. The clicking is from the bimetal strip that flexes due to electrical resistance heating up the strip and repeatedly breaks then connects again as the metal cools and makes contact. A blown fuse would not send electricity to the blinkers to cause the rapid blinking. The reason it blinks faster with a blown bulb is there's more voltage to the strip so it heats up faster and breaks the connection sooner but doesn't allow it to heat up enough to cause a longer off delay.
Not as much any more. Most vehicles now just play the sound over the driver's door speaker. The blinking is a PWM. Lack of current draw results in the PWM firing faster.