Need advices on an AC fan control circuit
1-) Is my non-inverting comparator good enough? I mean do I need some extra caps around any nodes? R2 and R3 sets the threshold, but are they okay in values? 


You need power supply decoupling capacitors, 100 nF, close to each chip. Those reference resistor values look fine.


  Do I need a pull-up resistor from the output to the Vcc?




Figure 1. LM324 innards showing both a pull-up and pull-down transistor.

A quick look at the LM324 datasheet shows that it has strong pull-up to V+ by Q6 and a strong pull-down to V- by Q13. It doesn't need a pull-up resistor. Note, by the way, the Q5-Q6 Darlington arrangement: this can never pull closer than two diode drops away from V+. The arrangement around Q13 is different and the datasheet says Large Output Voltage Swing 0 V to V +/−1.5 V.


  2-) I haven't experiment the final circuit and haven yet bought the relay. Do you have any suggestions regarding inrush currents, decoupling caps or any safety issues or anything related?


You'll need a transistor to drive the relay. The LM324 won't be able. Put a snubber diode across the relay coil to prevent switch-off voltage transients blowing the transistor.


  3-) And finally, is there a way to separate AC neutral and GND in LTSpice? You see in my circuit it doesn't represent the real life scenario since AC neutral is the same with GND in my schematics.


I have no idea.




  And finally the problem was that this type of fan and controller was not able to totally stop rotating unless someone manually switched it of from the controller's mains switch. 


Did you try adjusting the MIN speed pot on the controller? You may be able to set it low enough to stall the motor. Watch out for higher than expected stall current but I don't think you'll have a problem with a triac controller as at minimum speed it will be giving very brief pulses of low current to the motor.

Congratulations on getting it all to work.