Welcome!
So many things here, I'm not sure where to start. I can't give you specific responses to many of your questions, so I'm not going to go by the numbers, if you don't mind. Maybe if I had a similar type pond and lived closer. (Have you looked for pond and water feature clubs and organizations in your area?)
I would get a glass container, the taller the better, and pull a sample of your cloudy water out to see what it does. If it's truly silt in your water, it should settle out over time (24 hours, plus or minus a great deal depending on what's in the water) in an undisturbed, covered container. You can learn a lot about how long that takes, what the sediment layers look like after it does, and what remains in the water, and what color the water is after settling.
I would also do some water testing before adding or removing anything. You'll want to know pH and hardness for sure, but a full battery wouldn't hurt if you're thinking about investing in this pond. There should be local places that can help, and you can ask your county extension for resources.
It sounds like you've studied up on flocculants a bit, so that's good. I prefer agents that keep working over time, keeping new particles that enter the water column aggregated, to one-time solutions, but there's still some discovery needed to fully understand what is happening. You need to know pH and sediment composition before you know how things like alum and gypsum will work. Changes in pH and mineral composition are fairly permanent fixes, but lots of things can go wrong. In the ponds I've worked with, bacterial action is enough to bind particles together and clear the water, as long as everything is healthy and has enough oxygen. We have a fair bit of silt, but not much true clay, which may make a difference (see settling layers).
Oxygen can help tremendously, and everyone with a large pond around here has some kind of surface agitation. Some people just hook up a pump to a big pipe and let it shoot out over the water--new water surface area exposed to air over time is what matters, so turning over a large section of the water's surface helps. Bacteria and algae both benefit from this, and they make up the bottom of the food chain. A heavily planted pond likely won't need additional aeration.
And I would recommend some plants. Native ones, if you can figure that out (which means I can't help you much from here on the other side of the country) since non-native water vegetation that is vigorous enough to make a difference over a large pond quickly becomes problematic and invasive. Vegetation provides all the things you mentioned, but it also oxygenates and diversifies the ecosystem (for better and worse). Research is needed here, and I wish I knew more about what grows in your part of the country.
You mentioned walnut trees--I've seen trouble when walnut debris falls into the water, but if they're not overreaching they probably won't be a problem.
Really, 100 catfish in 1/2 an acre shouldn't be causing you any water problems. I don't know how accurate that number is, though. Given the size of the pond, I doubt the problem is the fish.
Hopefully some people here can give you more specifics, but I'd really start with finding out more about your sediment and your water chemistry.
