Electronic – What class of audio amplifier is this unusual configuration

Your schematic is squished too small so that part values and designators can't be read, so I'll only answer in broad terms.

For starters, the input section is missing a deliberate DC bias. You have AC coupled the input, which makes sense for a audio circuit, but added no deliberate DC bias of its own. Only the middle stage has a deliberate DC bias, but this is AC coupled to the output stage so that bias does not follow to the output.

This circuit also can't seem to make up its mind whether it is meant to be single supply or not. The three stages seem to be running from a single supply, but then the output is shown DC coupled to a load to ground, which makes no sense relative to the other parts.

To fix this mess, I'd start by adding some DC bias right after the input capacitor. That should float the first stage at some reasonably predictable operating point. Perhaps you can DC-couple the second and third stages, then use the DC of the output to bias the second stage. This feedback would be filtered to only effect the DC operating point. The AC feedback would still go back to the first stage as it does now. Beware of instability though.

This kind of output stage will have a little crossover distortion. You didn't give enough specs to determine whether that matters in this case. If this is meant for high quality "hi-fi" operation, then you need to deal with it. If it is meant for something less demanding, like a public address amplifier for example, then the overall feedback may deal with the crossover distortion well enough. It won't be large, and for many basic applications that will be good enough, which allows keeping things considerably simpler.

In any case, you need to add small resistors in series with the emitter of each output transistor, else you can get large currents thru both and thermal runaway.

Q1: A full class AB amp could amplify voltage and current. I suspect you mean a specific topology of a class AB amp. And from what you describe, I suspect you mean two transistors driven with diodes as voltage shifters which yes does not amplify voltage, only current. The example below is only Class B, but will suffice just fine for an alarm.

Push-Pull Class B Amp

Q2: If your expected circuit is anything like the picture above, then you shouldn't be using current from the uP to send directly into the BJT's. Your uP sends a voltage, this should be coupled by capacitors and then you should send the voltage signal to the amplifying stage which then uses current from the supply to drive the BJT amplification. Even if you choose not to capacitively-couple your input signal to your amp stage, you'd tie in the uP signal between the diodes. You'd notice that even then, any current coming from the uP is not current that travels through either BJT. Therefore, you don't have to worry about the gain between your uP output and the amp, you have to worry about the current flowing through R1 and R2 and how much gain you'll need there.

Q3: Becomes moot if what I answered in Q2 is correct. This is because your current input is determined by R1 and R2 then rather than the uP output.

You'll likely want to place an extra capacitor as well between your amp output and Rl (a much larger one than the input side to the amp) so that you isolate DC voltages from the AC signal you're trying to send to the speaker. This would allow your speaker to operate from -2.5V to 2.5V (for example) rather than from 0 to 5V which is what your uP will be powered/operated with.

EDIT:
For more detail on component values, then you'd want to look at BJT gains.

First you'll want to know what maximum voltage you'll want to drive your speaker with. Using Ohms law and the relation to power you can use V^2/R=P. Or V = sqrt(P*R). That would give you V=1.26V but that's only at DC. A sine wave with equivalent power to DC would be sqrt(2) times that or 1.79V. And since it's a sine wave it'd be +/-1.79V or 3.58Vpp. That's a pretty large range, but within the operating capability of a 5V power supply.

Using the other form of the power equation P=I^2*R, you can find the maximum current you'll want to drive the speaker with: 0.158 amps but once again that's at DC so your maximum current will be sqrt(2) times that for AC or 0.224 amps. Your Hfe plot shows that your current gain ranges from 70 to 170 so pick a number in there and see if it'll work. 100 is a good place to start. If you want the ability to output 0.224 amps, with a gain of 100, you'd need the input to the transistor to be 0.00224 or 2.24 miliAmps. I'd start there with your resistor biasing. Assume 0.7 V drops across each diode, Vcc is 5V and R1 and R2 are equal. That means 3.6/2 V or 1.8V drops across each resistor. 1.8V/0.00224A = 803 ohms. I'd start there and see if it works.

For the coupling capacitors I'd go as large as you have them within reason. 1uF to 100uF is probably a good place to start.

A word of caution, do not take the values I've given here and blindly place them into a production PCB. It is bound to fail miserably. Take the circuit and component values given here create the circuit on a breadboard and then toy with/modify it until you understand it prior to trying to put it into a PCB. All values here are estimates and first order guesses. Also, DO NOT use 12V or 19V to drive a dainty speaker like that unless you want to watch a lot of diodes, transistors, and speakers release all their magic smoke.