New Filter. Oxygen Problems? Dying Livestock

I had an over hanging filter for my 55g tank, but it was pretty loud so my dad bought a fluval. I asked him to grab a powerhead too, but he used to raise freshwater fish and therefore disliked them and didn't buy one. I set up the fluval-rinsed it all well-and started it. I noticed that it was kicking out a bunch of white dust, which I believe is from filter media.
This morning when I woke up, my coral beauty, large cleaner shrimp and emerald crabs were dead. My royal gramma was floating around. My green chromis, ocellaris clownfish pair and damselfish were all gasping but swimming around ok. My valentini puffer is turning a darker brown and has stuck herself to the tank wall like she's sleeping and Is gasping there. I can't lose her! I love her to pieces!
I did a 20% water change and stopped the filter. I put a different-smaller-overhang filter on for circulation, and I know that you shouldn't use airstones but I put two in.
Specific gravity- 1.023
pH- 8.0
Ammonia, nitrates, nitrites-0
 
My chromis, clowns and damsel are looking a tiny bit better (more lively), but my puffer is still a dark shade and gasping. Please help!!

You ought to know that removing the hanging filter you removed the only source of the tanks cycle.
 
Can I assume there is no live rock in the tank?

No, there's live rock and sand.

Hm, Okay what it sounds like. Is you removed the old filter, Holding a fair bit of the tanks cycle. Over night there was a spike of ammonia and your fish have suffered for it. You won't be able to trace it now as the live rock would have broken it down by now but I believe that was the case.
 
It might be a bit late to get the old media out the old filter to put in the new one, Do you have good surface movement anyway? Good ripples and such.

I do at the moment, because I put in a different overhang with no media in it at the moment. The old filter won't start.

Think its a bit late for the old filter, You should be fine with your set up now, Just keep watching ammonia. Ammonia damage causes rapid breathing.

The fish are calming down now. The poor puffer still looks awful...I really hope she makes it. My royal gramma also looks rough, but I'm guessing he'll be ok. I believe my Duncan coral is a goner. The zoanthids look normal though. Is there anything else I can do for the fish? I added more stress coat than usual while doing water changes.

Normally I would not run a filter on a SW tank, So am not entire sure myself. Still somewhat of a SW newbie but the ammonia etc is the same when it comes to FW.

I'm not sure if I should move her to a QT or not..I feel like moving her at this point would just stress her more. Ammonia is still at 0

If ammonia has been zero each time it was checked, assuming the kit is ok, it doesn't sound so much like a mini cycle as some sort of contamination. Crabs like emeralds can often withstand pretty bad cycling conditions over short periods of time, which is why they manage to hide out on live rock in curing bins, so the fact that the crabs are dead suggests to me it's not just a small cycle issue. There are two things that I always recommend doing whenever animals die mysteriously like this:
 
1. Get the surface jumping with airlines. The idea that airstones and open-ended airline shouldn't go in a marine tank is a myth. They can be a nuisance for causing salt spray with open-topped systems, but they are not detrimental to the vast majority of tanks (some exceptions where circulating bubbles need to be kept to a minimum - but most reefs and fowlrs do not have that issue). In fact, some people keep an open-ended piece of airline in tanks as a precaution, myself included. If you have gasping fish, the extra agitation from bubbles breaking the surface can be the difference between life and death. Not all gasping problems will be solved this way (such as toxin-induced ones), but it's worth a shot when the source of the problem has been nailed down.
 
2. If there is no skimmer on the tank, get a sock of activated carbon into the tank near a place of active flow. Again, not a sure fix, but better safe than sorry on skimmerless systems when a toxin has gotten involved. If there is a skimmer, check it and make sure it's performing correctly. Sometimes people will put a sock of carbon in during emergencies for the extra filtration even when a skimmer is present.
 

 I noticed that it was kicking out a bunch of white dust, which I believe is from filter media.

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What media was in there? Is it possible anything equipment-wise could have malfunctioned, overheated, etc.? 

i was going to say run some carbon incase of a pollutant of some sort but donya seems to have beat me to it lol

I'm not sure what the media is called. Small white clay looking tubes. I want to assume that it was an oxygen problem, because there was very little surface disturbance and my zoanthids and Duncan coral look perfectly fine. My royal gamma has fully recovered, and is hiding around my liverocks the way he always does. It turns out that I was mistaken-my two emerald crabs are fine. I'm still worried about the puffer, but she's hanging in there and appears to be breathing more. Also, the water is cloudy...ammonia, oxygen, media dust?
I'm also pleased to say that my two firefish, who I've had for two days, are alive and looking fine.
My coral beauty was the largest fish in the tank...can I assume that she and my large shrimp had higher oxygen needs?
Would pictures of things help figure this out?

Ok. Old filter, (the salt creep is bad. We got the entire system except livestock from a previous owner.) New filter and the media in question. My puffer, zoanthids and some of the other fishes.

These wouldn't fit in the other post...and here's a picture of them healthy for reference.
 

(the last two pictures are of them healthy.)

Update: my puffer swam across the tank!! She didn't take any plankton, which is her favorite food, but she did swim up to my face like she normally does. She stuck herself back in her sleeping spot after a couple of minutes. I'm very happy.

That media is usually referred to as ceramic rings. They get a lot of hate as a saltwater filtration medium, but won't be the cause of the problem as long as they were new and not carrying any organics with them. The main problem with them is that they are prone to crud accumulation over time unless the canister is prefiltered (something on the intake to stop large particles and uneaten food going into it), but the same problem can actually happen with live rock rubble or any other media in the absence of a prefilter and regular maintenance.
 
On a side note, the canister will not provide sufficient flow long-term. They're great for housing chemical media like phosphate remover and carbon, but since you said there are no powerheads/circulation pumps in there you should really look at putting one or two in soon to get flow lengthwise in the tank. The extra circulation is important for the rock to stay healthy. I also have a 55gal and have two canisters on it (one for rock and one for chemical m, but I also eventually had to add two power heads at opposite ends of the tank.
 

Also, the water is cloudy...ammonia, oxygen, media dust?

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Hard to tell sometimes unless you can see particulates. It could be media dust (harmless) but could also be a bacterial bloom in response to something. Bacterial blooms can happen when the biological filtration is upset (seems most likely here and would account for the O2 being sucked out), but they can also sometimes happen in response to contaminants.