You might want to follow the link in my signature to get some of my thoughts about space. I don't believe in magic square feet per chicken, there are way too many variables in climate, flock make-up, goals, management techniques, and so many other things that one size does not fit all.
What you have will probably work for your goal of one rooster and 5 or 6 hens. The size of your coop is really nice, it gives you some flexibility in managing many issues. For your ideal flock the run will work when it is available, but will it be available in your winters? If they are locked in the coop section only during a lot of winter you may need every inch of that coop space, though for 6 hens and a rooster it should work. I find the more I crowd them the more behavioral problems I have, the harder I have to work, and the less flexibility I have to deal with issues. I don't believe in shoehorning every chicken I can into the available space. I believe in deciding what you want and them providing them enough space. It is pretty likely you won't know exactly what you want until you get some experience so don't jump in too deep too quickly.
In the winter, even if it is covered with snow, you might want to give hem them the option to go out if they wish. They might surprise you.
Do you plan to add more chickens later, either having a broody hen hatch and raise some or brooding incubator chicks yourself? When you integrate those you may find your space pretty limited. We do it, it can be done, but during integration can bring out some space limitations of your set-up.
As far as keeping other species, you can do it, it is done. But I suggest you don't until you get some experience. Don't make it too complicated when you are starting out.
That chicken wire run will provide some protection from predators. It's true that certain large predators like a big boar raccoon or a large dog can tear chicken wire or even some hardware cloth, but it will stop a lot of predators. Being inside the dog run and with your dogs in there will provide additional layers of protection. I assume your dogs are there 24/7? They don't do you any good when they are locked in your house.
Practically any predator can and will hunt during the day, including the ones that "only hunt at night". Night is your biggest danger though. Predators are often more active around human habitation when humans are in bed and you don't have human activity to scare them away. The philosophy that has worked well for me is to provide a predator resistant place during the day and secure them in a predator safe area at night. No matter what you do no one can ever guarantee that you will not have a predator attack, you can only reduce the possibilities.
One weakness might be how that chicken wire is attached. I can't see that clearly. How easy would it be for a predator to get hold of it and rip out the connection? When I'm attaching chicken wire, hardware cloth, or 2x4 welded wire if I can I cover the ends with a board and screw it on to make that connection as secure as I can.
As long as it is closed that dog run should stop any non-climbing predator. As long as you trust your dogs I don't see any big benefit to adding heavier wire to that chicken wire around the lower section unless you cover the top also. In your case it appears to be mostly there to keep the chickens in. Personally I'd be comfortable keeping them in there during the day and locking them up securely at night. That does not mean it is impossible for something to go very wrong this afternoon, I just consider it unlikely.
Good luck!
