That is not so good.

> So the 3.3V power pin from the RPi is connected to the breadboard's top-most rail via a red jumper (note this isn't actually used by the rest of the circuit but I don't believe it harms anything, right?

Correct, it is pointless that way.

> When my software (running on the RPi) fires a signal to the buzzer via GPIO output pin, the buzzer will sound, etc.

Maybe. Also maybe you will fry the GPIO, since that buzzer looks a lot [like this one][1], which "[carries] a maximum **45mA**". The GPIOs (excluding power pins, which are not actually GPIOs) are rated at 15 mA. Although a lot of people will report doing okay at 20-25 mA, 45 is very likely too much.

You need a resistor to limit the current.

> pull up resistor vs. pull down resistor

You've misunderstood the purpose and circuitry of a pull-up/pull-down resistor. They aren't in line with the pin to protect it, they are in line with ground or a voltage source in order to prevent an input pin's value from floating in a [high impedance state][2]. Here's a pull-up circuit:

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![schematic](https://i.sstatic.net/FgKKh.png)

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Whatever the GPIO would be attached to would go bottom left. Notice that means the resistor is not in between it and whatever; it is between it and an internal 3.3V source. This is used to "pull-up" (i.e. to digital "high") when whatever it is attached to is in high impedance (i.e., connected neither to ground nor a voltage source). The purpose here is to keep the pin high until the whatever (say a button) pulls it to ground (i.e., digital "low") -- in which case you'd want a smaller resistor in line to prevent a short.

You are using the GPIO as an output, and need to protect against too much current draw. For 20 mA:

 3.3 / 0.020 = 165 Ohms

A bit bigger would be wise, but no smaller.

However, this does not take into account [the resistance of the buzzer itself][3], which may in fact be too much for this to work at all.


 [1]: https://www.radioshack.com/products/radioshack-6vdc-mini-buzzer
 [2]: https://en.wikipedia.org/wiki/High_impedance
 [3]: https://electronics.stackexchange.com/questions/205510/is-there-an-internal-resistivity-in-buzzers