Constant current load
1.What is the use of the transistor Q1? I understand that it's an emitter follower but wouldn't I achieve the same using a voltage follower from a spare op-amp?


The circuitry around Q1 set the initial maximum current reference voltage. They presumably used a dual op-amp so had no other device to use and a transistor was cheaper than adding another one.


  2.Are R4,R8 and C1 crucial for the readout or can I just omit them out


U1 provides a differential amplifier subtracting the current feedback from the required value, so yes R4,R8 and C1 are required.


  3.Could I use this for a DUT using AC?


No it will not work on AC.


  
  I built the circuit as is on breadboard but the current through R15 is not going over 100mA! Even when the reference voltage set at U1 is 5V which should give me anything close to 5A. The MOS-FET I'm using is a logic level FET according to this IRL3705N datasheet. I can confirm that the FET is original since it's from a reputable source and the op-amps are working fine in other constellations on the same breadboard. Maybe I'm overseeing something or the circuit is just baloney.
  


Are you sure R15 is 1 ohm... looks like it is supposed to me 0.01Ohms to me

as for baloney... the circuit could do with a little work.

Here is a simpler block diagram of your circuit.