Assuming the "peppered cory" is
Corydoras paleatus, they will need a larger tank before very long. To your questions.
1. Do cories breathe faster than other fish?
I noticed some heavier breathing than what I'm used to seeing. I did also notice them sifting sand through their gills, which was rapid. I checked ammonia and it was 0. Didn't check the rest. They are not going to the top real frequently either but I have noticed flashing. No white spots, no red gills. Love the pellets.
Respiration in all fish should be "normal," and it takes some observation of fish over time to discern "normal" and recognize "rapid." I don't know that I can accurately explain this; it is like when my non-aquarist friend once asked me how I knew how much food to feed each tank, I could not answer--it is one of those things you learn and instinctively know. I regularly observe the respiration of my fish, it is one of the first things I look at when I sit down in front of any one of my 8 tanks, something that has become second nature I guess. The respiration rate of the fish, and cories I find especially significant in this, can tell me if I need to clean my filter hoses; the build-up of organics in the hose and the increase in the organic sludge in the filter will slow the flow--not usually anywhere near enough to make it obvious to me, but the fish respiration or surfacing more is usually the clue. The fish have never been wrong yet.
TYo attempt to explain respiration...the gill covers will be moving sufficiently to see the movement (though sometimes you have to look very, very closely, it is so minimal) but not so much as to be what we term flaring, when the gill covers are extended out and the movement is more rapid. If you can see the gill filaments, the fish is respirating too rapidly.
However...faster respiration is normal when feeding. And when "playing" with each other, or when interacting to spawn. In the early morning, if the tank is planted, respiration will be more obvious because of the accumulation of CO2 during the night; I tend to increase the surface disturbance when I see this, and it tends to disappear or at least minimize. Any stress such as water changes or working in the tank can increase respiration. All of this is because the first thing a fish needs to do when anything is out of the norm is assimilate more oxygen to keep feeding the homeostasis.
Surface breaching to get a bubble of air is similar; the more often, the more likely something is wrong, but again the activities mentioned above can be related, so it is not necessarily a danger.
Ich will also increase respiration, and flashing is related. You may never see the spots, as the fish can, if free of stress, deal with this. It is only when fish are under more severe stress that they may not be able to shake this off. They flash because ich first attacks the gills, where we don't see it. I usually but not always see this in newly acquired fish in the QT. But it generally disappears after 2-3 days. I would certainly not jump into treatment (aside from avoiding further or unnecessary stress as much as possible); additives of any sort will always add stress to fish, and even increasing heat will do this, so you don't want to go down this route without very good reason.
2. Do the omega one shrimp pellets cloud the water?
It's not much, but a bit noticeable. I don't know if this is caused by over feeding possibly? I'm not sure how many pellets to give them.
I have used these for many years, and never do they cloud water. This is more likely a bacterial bloom, if fish are added to a fishless tank (assuming it was cycled, or has live plants--floating plants are always beneficial in a QT as it calms fish considerably). I wouldn't worry about this, but monitor ammonia and nitrite.
3. How many pellets should they get? Once a day or more?
First day I gave them 5 or 6. Then started more than that as they found they loved them. Only once in the morning so far.
I don't feed fish the first day they arrive, but the next day I usually do. With seven cories, I would put in 3-4 pellets at a feeding, and only once a day. I have 50 in my 70g, and today they happened to be fed these shrimp pellets, and I put in 8 or 9 in total. The pellets in my can of Omega One are about 4, maybe 5 ml in length.
Fish are best fed early in their day. Wait one hour after the tank light comes on, to allow the fish's eyes to become adjusted (it takes around 30 minutes, but best to give them one hour). It can be any time after that, but not later than an hour before the tank lights go out, though with cories that are nocturnal this is less of an issue, and they can be fed at night. If you feed fish at roughly the same time every day, they will quickly learn this time, and expect feeding. This is beneficial; I have rarely fed them outside the "norm," and it is interesting that they should little if any interest--simply because they do not expect food at this time. Fish learn day/night times, it is part of their circadian rhythm, which is why regular light on/off times are so important.
4. Should I feed them anything else other than these shrimp pellets?
Yes, just for variety. I use the shrimp pellets as the only food for new cories in the QT, and for fry I rescue from the canister filter until they are large enough to go in the main tank. This is nutritious food, and Omega One is one of the most reliable brands, with no "meal" but whole fish, shrimp, etc.
In the main tank, once the fish are settled, I alternate three sinking foods for cories (and loaches and whiptails and twig catfish). Shrimp pellets, Omega One Veggie Rounds, and Nutrafin tabs (these have earthworm). Once a week, on the water change day as their treat about two hours after the WC, they get frozen daphnia and bloodworms. One day a week is fasting. The Veggie Rounds are important for giving them vegetable matter which benefits a healthier digestive system. There is fish in them too. My substrate fish love all of these; the cories will be clustered around a disk or tab for several hours.