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.

You are getting a lot of good advice here and in my opinion I don't think any of it is bad. In the end you have to sort through everything and decide for yourself what will work best for you. There are many ways to implement a subwoofer or distributed array. Unfortunately, in each of these threads there is one particular member that shows up with their it's my way or the highway monologue. This one is no different.

Let's get to the point. You have $1000 for your budget if I read correctly. That's not going to get you very far in the DBA religion. Heck, my DBA cost $4k and I built my own speakers. Unless you increase your budget you are better off searching the forums for information on good powered subwoofers, then search the listings to see if you can find 2 or 3 used ones that fit within your budget. At a minimum you will get an idea on what multiple subs can do for your system, as well as learn the ins and outs of setting them up which is critical.

If you did have a $3k or so budget I'd point you to Duke Lejeune of Audiokinesis. His turnkey DBA is the most thought through commercially available solution. More importantly though, I think you should look him up and contact him directly, as he is a wealth of information on this topic and speaker building in general.

All of my subs are on the floor, with two at the back wall between the speakers and facing each other, about 3' apart. The others are spread out along various parts of the wall and facing toward the room. I am trying to figure out a way to get one or more of mine in the air but in my room it isn't easy. I might need to build some pedestals.

I have also thought about trying one or more subs in a down firing position, but would be curious as to how far from the floor the down facing woofer should be. Same would be true about the woofers facing the wall.