Let's try this stepwise:
The thermistor is undefined. The op amp acts as a comparator. When the voltage across the thermistor is above the setpoint voltage on the 47K pot (has to be wrong, potentiometer values are 50K) the output of the op amp goes high, turns on the transistor, and that puts current through the relay coil.
The diode is a suppressor to prevent reverse voltage from the relay coil damaging the teansistor at turn off.
Provided the relay gets enough current, changing to a 9 volt supply will not affect circuit operation - unless it's a battery. Then battery life will limit operation.
A 358 is as good as the 741 for this circuit.
I am at a loss to understand the last part. If you are going to use 9 volts, why is the voltage on the transistor collector only 1.2? What kind of relay is going to operate on that voltage?