Hmmmm... how would having random voltages ever be useful?
RNG. Other than that niche use probably never.
Why don't all the pins have built-in pullup resistors to "quiet down" the voltage when there is no signal present?
Cost? Die space? Didn't feel like it? Who knows?
Some micros have built in pull ups and pull downs. I think the nRF52 series has these. It all depends on what the designer wants to put in.
Right! I've never used an external pull-up with the Arduino. (There may be cases where you need a different value resistor, or you may need a pull-down.)
You've just never needed to. Some projects do need it.
I have digital pins controlling a relay. Is it possible for random environmental fluctuations to cause a digital pin to output a high voltage and cause the relay to inadvertently actuate? Would it be necessary to add a resistor to make sure the voltage is always at 0V unless commanded to output?
It depends on the characteristics of the input.
It is high impedance inputs that are sensitive to noise. A pullup resistor doesn't just offer a connection to a defined voltage, it also lowers the impedance of the input by putting the resistance in parallel. CMOS digital inputs have very high input impedance (hundreds of megaohms or more, usually), and are very sensitive to noise. Lowering the impedance means that you need stronger noise to affect the input.
For many things, the roughly 60k pullup resistor built into the AVRs is adequate, but there are times when it's not. If you are in a high noise environment (such are near mains cables, an engine or motor, or running really long cables to the switch), you need lower impedance.
Most cheap relay modules will probably just have a BJT or optocoupler input. These are basically diodes, which are current driven and very low impedance. There's not much danger of any kind of noise tripping up one of those. Any noise strong enough to activate an optocoupler is probably strong enough to cause you many more serious problems.
If the module has a CMOS input buffer, you might need to be careful about that.