You are likely to get various opinion some this one. Some people do get by without a run. I let mine free range but find that a run comes in very handy at different times.
From your description of your fences, I'm guessing you live in an built-up area, probably a subdivision. Cats are normally not a threat to grown chickens, although they will sometimes take chicks. In additiona to opposums, raccoons and hawks, I'm sure you have many other threats. Foxes and coyotes come to mind, even in subdivisions at night and even with those fences, but especially in suburbia dogs are a huge risk. Foxes, raccoons, opposums, and others will climb those fences like they are not even there. Coyotes and many dogs can easily jump a 4 foot fence. And your chickens can fly over those fences if they want to. What often happens is that they fly up to perch on top and who can guess which side they decide to hop down on. But if properly motivated, they can clear those fences. Many people keep chickens in with those types of fences without a problem, but you will find with chickens that they are often not consistent. What works for one person may not work for another.
Any time you free range, you take the chance that you will lose some to a predator. This does not mean that every predator in the county will instant head your way for a feast. That does happen to some people, but many of us go a long time in between predator attacks. I can't tell you when a predator will attack, but I can tell you that some of us are at more risk than others.
You can get by without a run. Many people do. Whether you should is the question.
My philosophy is to let them free range but lock them very securely in a coop at night. I have had losses. Until someone dropped a couple of dogs off in the country instead of taking them to the free drop off animal shelter, I had lost very few using this method.
Although I usually free range them, I find having a run comes in handy in many different cases. Sometimes I have to retrain one to use the coop nesting boxes. If I see a predator in the neighborhood, I leave them locked in the run. Maybe if I need to work with them, banding them or medicating them. Sometimes it is handy to leave some locked up when I integrate. The run comes in very handy and gives much more flexibility in how I can handle them.
Will a rooster protect your flock? It depends on the rooster. A good rooster will sacrifice his life to keep his flock safe and will be on the watch for predators, especially hawks. But when danger is present, my current dominant rooster will lead his hens to safety instead of positioning hiimself between them and the risk as he should. Sometimes the dominant hen will assume many of the protection roles of a rooster. Sometimes but not always. A good rooster will make your flock safer but there is no guarantee any specific rooster is going to be good. And no rooster is going to regularly stop many big predators, but he can sometimes slow them down.
