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, 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. Here's a pull-up circuit:

^{simulate this circuit – Schematic created using CircuitLab}

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, which may in fact be too much for this to work at all.

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, 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 probably need a resistor to limit the current, at least initially -- the buzzer itself will have resistance as well. Realistically it looks a bit beefy to be controlled this way. You would be better off feeding it 5V toggled with a transistor.

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. Here's a pull-up circuit:

^{simulate this circuit – Schematic created using CircuitLab}

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, which may in fact be too much for this to work at all.

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, 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. Here's a pull-up circuit:

^{simulate this circuit – Schematic created using CircuitLab}

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 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 (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.

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, 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. Here's a pull-up circuit:

^{simulate this circuit – Schematic created using CircuitLab}

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, which may in fact be too much for this to work at all.