Which order to apply room correction tools?

I agree you should first do everything without applying any filtering. Place the speakers, then find good locations for the subwoofers. Since you have 3 subwoofers, you are probably going to have to treat at least two of them as a single channel so keep that in mind during placement: in other words two of them probably need to be the same distance from your seating position.

Room treatments should go on either as part of your first step in speaker positioning, or right afterwards. Since everything coming next is going to be influenced by the treatments.

Second I would use Room EQ Wizard and Multi-Sub Optimizer to perform PEQ independently on each subwoofer (two of which will be grouped into a single channel in the SHD) to flatten their response.

Third, use Room EQ Wizard to figure out the appropriate delays (positive or negative) for your subwoofers, relative to one of your main speakers. Enter these delays into the SHD for the subwoofer outputs.

At this point, your subwoofers should have relatively flat frequency responses and be time-aligned with one of your main speakers.

Finally run Dirac.

@alanhuth I haven't used MSO myself but I believe it can be used on its own to determine the PEQ you should apply to the subwoofers. I would not apply any manual PEQ to the mains, and instead let Dirac take care of it. But you need to perform the subwoofer PEQ and time-alignment first, otherwise Dirac's 2-channel measurement and filtering won't work as well.

The time-alignment I am referring to is the same as what a surround sound receiver or processor typically does. It doesn't have anything to do with PEQ.

Imagine your subwoofer is located 100 feet away from your head, while your left speaker is located 10 feet away. If you play a continuous sine wave at your crossover point of 80Hz, you could adjust the phase of your subwoofer until there is no destructive interference. But when you played music, things would sound very wrong because sound from the subwoofer would arrive way too late compared to sound from your left speaker.

This problem will then present itself when performing the Dirac measurements, and consequently the computed filters will not really do what Dirac thinks they will do, and things will still sound wrong.

miniDSP has some articles about measuring the time differences between speakers:

• Measuring time delay for home theater with the UMIK-1 or UMIK-2 and REW
  
• Time-aligning speaker drivers with UMIK-1 and UMIK-2
  

I'm also assuming your left and right speakers are essentially identical distance from your head, so Dirac or REW will measure a very small difference between their impulse arrival times.