Q1:
Compensation means designing a filter network that is to be added to a circuit that is unstable to make it stable. It is very common in control circuits.
Q2:
Strictly speaking it is not a pure integrator, but sometimes it will more casually be called an integrator. A pure integrator has transfer function 1/s while this has 1/(a*s+1) or similar where you note the denomoniator has a constant term added. The main difference is we get a 90 degree phase shift with an integrator, but with this network we dont get 90 degrees but less.
Q3:
You'll note that if you compute the transfer function Z2/Z1 for the part with the R's and C's you'll see that when we eliminate some parts we get a high pass filter and when we eliminate other parts we get a low pass filter. Thus when we make some parts larger or smaller we fade in or out those parts and the character takes on more one or the other. It's also possible to adjust so that they cancel out.
Q4:
It is entirely possible to design a network using all inductors and resistors or combination of inductors and capacitors and resistors, but it's certainly cheaper in most cases to use just capacitors and resistors. It's even cheaper not to have to use op amps, but in many control circuits we already have an op amp and so we are really adding components to a circuit that already exists and so already has an op amp, so we design the network to work with that rather than a stand alone RC or LCR or whatever which has input and output impedance issues.