If Corey is a Pictus catfish, then he may eat small fish as he grows. If their sizes are similar, he could go in with the oscar and Jack Dempsey. This site says they grow 6-10" (they list two kinds) and will eat fish under 2.5" long:
http://www.mongabay.com/fish/pimelodidae.htm (search for Pictus for the correct section). I actually got one of those years ago from a store before I knew much about fish and put it with goldfish (they said they got along). The fish shredded the goldfish fins immediately. I put him in the bathtub overnight to return; the tub drained overnight (hey, I was a dumb kid!); the stranded catfish was alive in the morning but died before I could get back to the store.
"Is Corey a type of catfish that will eventually eat his tankmates?"
Yes.
"How large would a fish have to be in order to be safe in the same tank?"
At least 2" long. Danios are smaller than that.
"if my danio might have an infection, can i still leave her in the same tank?"
That's up to you.
"Did she already infect my other fish?"
Yes, they're already exposed.
"Shouldn't it be harder for differant types of fish, such as the gourami, to catch an illness from the danio?"
Sometimes certain problems are more common in one species than another. My 14" pleco has never shown any problems even when he was a baby in a tank full of ick and now in a tank with danios with severe tuberculosis.
"I was looking really closely at my tank, and I noticed that the danio wasn't staying at the bottom all the time, just a little, then going back to the middle and top. In fact, i twas only yesterday that she was at the bottom mostly, maybe i just saw her when she was bored with swimming?"
Or, maybe when tired. Healthy danios spaz all over and rarely rest on the bottom.
"The filter I have is for up to thirty gallons of water. Wouldn't that keep the water in a 10 gallon clean enough?"
If the tank weren't overcrowded, and the filter's of good quality and kept clean, and you do water changes as well, yes it would.
"I know that sounds nasty, but i'm thinking if the water is clean enough for me drink without ending up in the emergency room, it's safe for my fish. "
Not necessarily. Instead of doing that, bleach the tank to disinfect it. Fill it with water and a few cups of bleach. Let it sit for a while (at least a few hours but a day is better). Then, drain that and refill with fresh water and a double dose of dechlorinator. Let it sit again for an equal amount of time (a day is best). Then, get rid of all that water again. Add rinsed gravel, refill, set up the filters and heater, aerate, add additives (dechlorinator, salt, etc.), etc. and add the fish a day later. See
http://www.fishpondinfo.com/fcare2.htm#tear for more details.
"Oh, and i'm going to leave the tank set up for six hours, with the filter running and everything before i let my fish in. Any advice on how to make this tank change any safer?"
Let it run for a day if you can before adding fish.
"Once i put my fish in the 55 gallon tank, is it possible to add new fish (without over crowding), after a while? How large can they be?"
Yes. I would wait at least a month. Stick to fish of similar size.
"Can I let my oscar and Jack Dempsey visit with these fish? I used to let my betta and gourami do that, they seemed to like each other if they didn't have to live together."
I don't think that's a good idea. Not only will the oscar and Jack Dempsey eventually want to eat those smaller fish, the stress of moving around and chance of disease transfer is not a good idea.