Electronic – Totem Pole Output Driver

When not to use totem-poles:

1. You can't use totem-poles for wired AND (often called wired OR, but most often they're ANDs). If one output is high and the other low you get a short. Always use open collector/open drain drivers for wired AND.
2. TTL totem-poles are highly asymmetrical: they can hardly source current, typically 0.4mA versus 16mA when sinking. So don't use them when you need to both source and sink current. CMOS totem-poles are more symmetrical, and don't suffer from this.
3. When the load you're driving goes beyond \$V_{CC}\$, like for instance a low-power relay which needs +12V on a TTL output. Sinking current the output will be low as expected, but when the output is high it will see the +12V, which will damage the part.
4. When the input you connect to uses a wildly different supply voltage. If for example the output would switch between 0 and +1.5V it wouldn't be directly usable as input on a 15V system. By using an open collector the pull-up can be connected to the +15V, and the output levels would be automagically according to that system. Important: Many new low-voltage logic families seem to have clamping diodes on their outputs, even open drain, so that you can't go beyond \$V_{DD}\$. I guess it has to do with the low-voltage process, but it does take away the open drain's most interesting feature.

I'll try to cover the parts of your question separately.

Multiplexing

Is the multiplexer represented in the schematic where it says "buffers"?

Nope! The multiplexer is the select digit part of the circuit. As you said, a multiplexer is an electrical switch: if I have \$n\$ "selector" inputs, I can choose from \$2^n\$ outputs. In your case, you have a two bit counter (because 1 bit isn't enough to count up to 2) which is connected to the "select" part of the multiplexer. The mux then sets one of its four outputs high, depending on what the counter is. If you make your counter reset as soon as it hits 3, then your multiplexer will set 0 high, then 1, then 2, and repeat this loop forever.

If the ultimate goal is to display the output on two additional displays, why couldn't we just wire the all the a's, b's, c's, etc. together?

When the mux has set digit 0 high, we only want to light up display 0 (and likewise for 1 and 2). If you wire the displays together, you can't control them all with different digits.

High End Digit Driver

The purpose of the transistor drivers are to supply/sink current to/from the LEDs, but it's unclear to what it's connected to electrically in terms of the emitter, collector, and base.

Look at a single digit driver. When its mux output is high, you want current to flow from your power supply into the LEDs; when the mux is low, you want to block that current. That means your which digit? output is probably connected to the base of the transistor, and setting it high allows current to flow from the collector to the emitter. Is that enough of a step in the right direction?

If the drivers are simply sourcing or sinking current, why are they connected to the multiplexer?

You'll have three drivers. You only want to turn on one at a time, and the multiplexer picks which one. They aren't "simply sourcing current", I guess - they're current sources that you can selectively turn on and off.

Counter Circuit

However, what is considered a desirable output here?

You want to count

0, 1, 2, 0, 1, 2, ...

so you'll need two wires (like I mentioned above). What parts do you have access to? What kind of counter circuits have you seen before?

As Nedd mentioned, you have a good oscillator set up - that'll be the input to your counter. Flip-flops would be a standard approach from there.