Okay I would add get aquarium salt and add it to the tank by the instructions on the back. Or you could use Melafix.

After he is all healthy you have to get a bigger tank. Goldfish grow to 12 inches and live 10 years and cant live with tropical fish! Your black moor and the rest of your goldfish will need to be seperated to a new tank that is at least 30 gallons. But your tropical fish and sucker fish can live in the 10 gallon tank.

Hi boogah and welcome to the forum, (firstly let me warn you this is rather long, so please take your time)

What type of filter do you have? Is it an internal filter or external filter?
It can be possible to cover the inlet on some filters, to lower the current, certain sponges can be used or covering the inlet with a material such as the foot end of a tight, these tactics are used in shrimp tanks and fry tanks, to stop them from being sucked in to the filter, so could help out with your situation.

How badly damaged are the fins? Could you describe it in further detail? Do you have a filter in the hospital tank?
Sometimes depending on the damage, fish can heal themselves as long as the water quality is good. You will want to keep an eye on things and make sure no secondary infections occur, such as fin rot.

When I say 'good water quality', you may wonder what that means, so I'll explain briefly.
In an aquarium there are 3 main elements that have to be considered; Ammonia, Nitrite and Nitrate.
When you first set up a tank, there is a process called 'cycling', simply where bacteria develops in the tank and filter, so that the harmful substances of ammonia is converted to nitrite and then more develop to convert the nitrite into nitrate.
So in a cycling tank you see ammonia levels rise, and then begin to fall, followed by nitrite levels rising and then beginning to fall, which is lastly followed by nitrate levels which begin to rise.
Ammonia and Nitrite are extremely bad news for fish and can cause illness as well as death, if not taken care of.
Nitrates however aren't as bad, as long as the concentration is below 40ppm. Weekly water changes, keep's the nitrates under control.

So as fish keepers, we use test kits to monitor these elements and re act accordingly. Ideally in a fully cycled system Ammonia and Nitrites will be 0ppm and Nitrates will be below 40ppm.

As Fantailer has said, tropical fish -depending on what they are- generally aren't compatible. Often tropical fish have different requirements in terms of PH and temperature, do you know what breed of fish they are? That would be helpful

Also when you describe your other fish as 'sucker cleaning fish' I am going to assume you mean a pleco of some kind. Again knowing the breed would be helpful, sometimes pleco's can be kept with goldfish, but there have been cases of the pleco sucking the slime coat from a goldfish. I'm not sure how common this situation can be, but it's something to be aware of. And also depending on the breed, will affect it's suitability for your 10 gallon tank, as some pleco's can grow to huge sizes.

May I also ask what type of goldfish the other two are?
Are they fantails or another fancy goldfish type, OR are they single tailed such as a common goldfish or shubunkin?

If you're not sure, you could always look things up on google images

In general for goldfish (as fantailer has already said) they grow pretty big and are extremely messy fish, excreting a lot of waste through poop and their gills give some off as well. As a result they need a lot of tank space.
The minimum stocking rule for fancy goldfish types, is 20 gallons for the first fish and then 10 gallons for each additional fish. So assuming your other two goldfish are fancy, you would need a 40 gallon tank, minimum. To successfully keep the goldfish.

If the other two goldfish are single tailed, then it's an entirely different situation.
As single tailed and fancy goldfish should never be mixed, as the single tailed fish are much quicker and can out compete the fancy varieties for food as well as potentially stressing them out, by being such agile and quick movers.
As a result of their active behaviour, single tailed fish are more suited to large aquaria (with a length being a minimum of 6 foot) or even better, a decent sized pond of around 1000 litres or more.

So it all depends on what types of fish you have

If you have any further questions, please feel free to ask, how is the Moor doing?
Is he eating? Are there any more symptoms physically you can see asides from the fins? Or is he acting oddly, such as bottom sitting?

I feel like I can see it in his eyes that he is sad... 

I am using Tetramin tropical flakes.  they are all floating on top... is there something safe i can use to sink the flakes with?
Also, I mean I cannot tell if his gills are hurt..it seems to me as though he has always had that kind of bottom sitting opening mouth a lot looking in the rocks for food kinda personality, but i have only had him for 1 week.

so priority- getting the food near him....any ideas?

oh darn, well it appears to me even with the flakes floating towards the bottom boogah isn't even attempting to eat them and just chilling on the bottom of the tank by a pebble.


my other fish in the 10 gal tank were swimming to the top a lot maybe for oxygen so i turned the filter on. but I am so scared it will damage my other fish...

Boogah's left gill is opening farther than the right.  The left side of his body if where the filter trapped him.

ON IT! wow you have been so helpful. i want to thank you again

Oh sorry, I didn't see that bit about lungs XD Thanks for picking up on that Noss!

If you can't afford a tank, ypou can buy a large plastic storage box

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One more thing....so this tank has a bottom filter I lined it in a thin nylon I had.  I dont have plants or very many soft rocks to line it with.....what if he gets caught in the suction of this filter too?? should i take some gravel rocks out of the other aquarium to weigh down the plastic filter lining the bottom?

An undergravel filter (i assume that's what you're talking about) is awfully weak, there is no way the goldfish would get caught in this, and if he would, he would most likely be dead or very close to dying either way.

Try looking for used tanks! 30 and 40 gallon tanks are pretty common, it's possible you could find someone that even wants to give it away for free If you wouldn't, sad truth is that you'll have to watch your fish get sucked into the filters, becoming sick from parasites and bacterial infections and dying one by one because of the poor living conditions.