Hi,
This kind of circuit can be deceiving. The op amp output is like a voltage source and it is driving the base emitter diode which because of diode voltage to current response and added transistor action produces a relatively high current for a very small change in base emitter voltage which then produces an output voltage across a relatively high impedance of 5k Ohms. So we go from tiny voltage to large current and then back to voltage which might be large until the op amp can detect that the output is too high, which of course takes time, and after it detects that it cuts back and then it needs time for it to detect the lower voltage.
The op amp itself has to be unity gain stable because the overall gain is 1 just as in a circuit that would regulate the voltage across the 5k resistor.
The huge nonlinearity of the transistor action can cause problems because of that voltage to current to voltage translation that occurs mostly because of that base emitter diode, so to help the situation a base resistor can be used. That then reduces the non linearity of the transistor and so help to stabilize. Maybe 100 ohms to 1k or so. That way the op amp has to do a little more work to get the transistor output to respond, and the transistor gets more linear so the voltage to current relationship relaxes a little which helps reduce abrupt changes that the op amp has trouble following.
To give a quick example, say the op amp output has to change by 0.010 volts and it does so, this could cause some milliamps increase through the transistor base, but with a base resistor of only 10 ohms that same 10mv only can cause at most 1ma increase and if more than that is needed the output of the op amp has to increase more, and that means a smoother transition from one level to the next.
As for the gain of the circuit, an input of 1v produces 1v across the 5k resistor, and that is 1/5000 amps which is 200e-6 amps. The usual approximation is Ic=Ie so the current in the collector is taken to be the same.
So the gain is 200e-6/1=200e-6 which is current/voltage which is a transconductance measure but is also referred to as a gain.
If you want to get more picky about the gain then if the Beta is 100 then the base current is 2ua and so then the emitter current is 202ua and the collector current is 200ua so the transconductance is still 200e-6, and given 10mv op amp input offset just to keep it simple.