Simple version of an internal op-amp circuit
The following answer concerns the principle operation only. That means, I do not speak about resistor values and alternatives to optimize the circuit.

As you have mentioned - T3 acts as a &quot;level shifter&quot; (with amplification). For this purpose, T3 must be a pnp type. Otherwise, the DC quiescent collector potential of T3 could not be lower than the potential at the T1 collecor. But this is necessary because of the desired DC output voltage of 0 V.

Quote: &quot;...T3 creates an inverted voltage on its collector (output), which is proportional to that current. (I'm not exactly sure as T3 may also use the voltage from T1 collector?)....&quot;
This point touches the general question if the BJT is to be considered as a current-controlled or a voltage-controlled device. In this circuit, and for calculation purposes, this question is not too important - both models can be used. However, if the current gains of the transistors are not known, the voltage-control model is to be preferred (using the transconductance gm=Ic/Vt).


UPDATE: I have roughly calculated the overall gain (using your resistor values and assuming a beta=200). The current through R3 is app I3=125µamp and the current through R4 is app. I4=2mA. The gain of the first stage at the base of T3 (with consideration of the input resistance at the base of T3) is app. A1=-65 and the gain of T3 is app. A3=-2. Hence, the overall gain is app. A=+130.