Which is better for a DBA (Swarm); powered subs or unpowered?

Is it possible to run 4 passive subs off of one amplifier, say like a Crown XLS series? Run in series or parallel?

I don't see why not. It appears the power ratings spec down to a 2 ohm load. If you run two 8 ohm speakers in parallel on each channel, or two 4 ohm speakers in series/parallel per channel you will attain a 4 ohm load.
I use an Acoustat TNT-200 to power my passive sub boxes. I have four 8 ohm speakers per channel in series parallel. A total of 8 subs in the system.

My set up is a bit different in that I triamp using a 3-way active crossover. My two-way ribbon speakers use an external passive crossover and I replaced it with the active. Under 100 Hz to the subs, 100 - 650 Hz to the bass/mid ribbon, and over 650 Hz to the mid/treble ribbon. A push-pull tube amp for the bass/mids and an OTL tube amp for the mid/treble.

Remember, 2 properly positioned subs will provide very good bass at your designated listening position only, but not throughout your entire room.

DBA subs give a time incoherent mess everywhere.

The Dayton has no delay so there is time smearing. What you end up with is a fat, detail-less low end pillow.

For music, two subs correctly positioned relative to the mains for the main listening position are the most realistic.

Of course, if the mains are an incoherent mess spraying sound all over the room, who cares?

Do you not have to adjust the phase of each sub independently? I wouldn’t think that with four subs in different locations, possibly pointing in different directions, that you would want the same phase or even the same volume levels across all of them equally...

I gather that with the dayton amp, you aren’t able to adjust each one independent from the other (?)  

ieales"DBA subs give a time incoherent mess everywhere."

Of course this is actually valid, verifiable, and repeatable how ever there are "lead heads" who love lots and lots of bass they like to "pump it up" so of course for them they need many sub woofers and for them it is best to have a variety of them mixing in 12 inch subs with 15 inch subs with smaller subs and a mix of low quality amplifiers often "Class D" which is really "Class Deficient"!

How loud do you like to listen? Do you listen to a lot of rock and want gut thumping low end or do you prefer classical or jazz and are looking to just fill in the bottom end?

With powered subs you need proximity to AC and you will need interconnects, unless your subs have wireless calability. If you are using RCA connectors and have some long runs over 10' to 15' that could be an issue. If your system is balanced then long IC runs should be fine. With passive subs you will need speaker cable and if you require big power and a long run then you may need to use heavier gauge wire up to 12 awg maybe even 10 awg.

If you like it loud, a good powered sub will offer better speaker protection. With passive subs and a separate amp you will more than likely get none, so if you hard clip your amp it will probably be re-cone city. The rule of thumb (at least it use to be) is to use an amp that is capable of twice the output power of the speakers maximum handling capacity. So if your subs can handle 200 watts @8ohms then you want an amp that can deliver 400 watts @8ohms and so on.

With powered subs you will have more individual control, which for some will be a good thing and for others not so much. A dba system itself makes integrating easier but as the room gets larger control becomes more important. It will be dependent on how sensitive you are to time domaine issues. You should be fine either way with your room.

Some subs (I think REL is one) allow you to wire them in a way that reduces the burden on the main amp to cover frequencies below a set cutoff, like 80Hz. Does anyone know if that is possible with the Dayton?

Almost all powered subs and I believe this is also true of the Dayton amp, only come equipped with a low pass filter, which allows you to set the frequency cutoff point of the signal that is sent to the subs.  You would need an actual crossover to do reduce the burden on your main amp. 

is it advisable to use all the same brand/model of subs in the swarm, or does it not matter?

There is not a lot of detail when you get below 80hz so in my opinion the subs do not need to match. I use three different brands of subs in my 4 sub dba with no issues.

what is REW? 

I believe REW is a type or brand of room analysis software.

Is it possible to run 4 passive subs off of one amplifier, say like a Crown XLS series? 

Yes as long it has enough power and can drive the impedance load that the speakers are presenting. The load will vary depending upon how the cabinets are interconnected.