You could successfully build a audio amp from many different types of BJTs. It will be the circuit, not the transistor, that makes the amp work well. I'd pick jellybean parts like the 2N4401 (NPN) and 2N4403 (PNP) and stick with them for everything except for the final power output transistors. Lots of parts could fill that role. If you have your own favorite jellybean small signal transistors, use them if you prefer. The ones I mentioned have reasonable gain and can handle up to 40 V, which should be plenty good enough to allow for a amp to impress your profesor with.
There are lots of possible power transistors to use as the final output. If you are aiming for a few Watts, I'd probably go with basic parts like the TIP41 (NPN) and TIP42 (PNP).
Again though, it's not the choice of transistor that will make or break this project. You can certainly create a impressive audio amp with the transistors I mention, but you can also make a mess. It's really up to the design. In audio, overall noise and harmonic distortion are high priorities. Those come from careful circuit design and attention to these parameters at every step along the way.
You can also use other types of transistors, like JFETs or MOSFETs. Those would require a different circuit topology to utilize properly, but can be used to make a good amp too. Since you will be going over BJT details more thoroughly, I'd stick to them for now. This will be a great learning exercise. Designing a amp with very low noise and very low distortion is not trivial.
I will try to give you some hints:
1.) Of course, you can define the output of the diff. pair at one collector node only (Vo1 or Vo2). It is just another option to use the other collector node as well and defining the amplifier output as Vo2-Vo1.
2.) Differential mode: Assuming linear operation (and this is always assumed) the current increase of the left BJT is equal to the corresponding current reduction of the right BJT. Hence, there is no current change in the common emitter resistor Re. Hence, you can treat each transistor stage as a simple common emitter stage without any emitter degeneration (the common emitter nodes behave as if they were signal grounded). The gain formula for such a simple arrangement is known.
3.) Common mode: Again, treat the BJTs as common emitter stages - however, now with emitter degeneration. Both BJT`s amplify the same signal. Again, the gain formula for a simple common emitter stage with Re feedback (degeneration) is known and can be used - however, you have to consider that the current change through Re is doubled because this resistor is common to both transistors. As a consequence, the feedback effect is doubled and the effect of resistor Re must appear as 2*Re in the corresponding formula.
4.) If you like to start with your small-signal model (and finding the gain values by yourself) the procedure is simple: For differential mode you only need to consider one of the transistors (because they are not connected (emitter signal grounded). This applies also to the common mode - however, in the gain formula you must replace Re with 2Re.
5.) Finally, an important comment: You have defined the diff. mode as V1=Vd/2 and V2=-Vd/2. This is OK. However, please note that in the first part of your question the expression for Ad does NOT apply to your definition. The given gain is for V1=Vd and V2=0. This is a non-symmetrical operation of the circuit, which is allowed but involves also a certain common mode voltage.
Best Answer
Given your instructor's remark, I suspect that you were actually told about "debouncing", which a very useful function. However, the circuit shown is a bad, bad example of how to do it.
Let's think about a switch making contact. Inside, you have two pieces of metal making contact, and on a timescale of milliseconds the two pieces can actually bounce, perhaps repeatedly, before settling down. If this happens the switch output will apparently show multiple activations where only one was intended. To combat this, a circuit like this can be used:
simulate this circuit – Schematic created using CircuitLab
You've noticed that it has the same general outline as your circuit, but a few more resistors.
Let's say that the switch is open, and gets briefly closed, then open, then closed for good (a single bounce). Current through R1 turns on Q1 and pulls the collector low. This also discharges C1. As a result Q2 is turned off and Vout goes high. When the switch bounces, Q1 is briefly turned off and its collector rises. However, the resistor charges up much more slowly than it discharges (for appropriate resistor values of R2 and R3), so Q2 never turns back on. As a result, the output Vout has been "debounced", and R2/R3/C1 can be selected for any desired bounce time to be ignored.
The original circuit is not very good, since the capacitor voltage swing is quite small, due to the clamping effect of the Q2 base-emitter junction.