I have pretty much the same set up - except you didn't tell me if your yard is fenced or not. You have all the room you need.
You can make a chicken coop for close to no money at all. You can get palettes for free and make them into a coop, or lumber for free on craig's list. If your dogs would bother the chickens then you would need to make the chickens a run or put up a run for the dogs - that would cost something ... that is if you didn't get those materials for free on craig list too.
Each year here in Austin we have what is called the Funky Chicken Coop tour. You wouldn't believe what people have made chicken coops out of! It's wild. Most of the people I talk to have built their own coops out of scraps they have around.
My husband took an old frame from a cheap metal greenhouse I had that didn't work very well as a greenhouse that used to have plastic on it and made that into a chicken coop - but my first coop for my 3 leghorns I bought from someone for 30 bucks.
With just 3 or 4 chickens you would be surprised how easy and cheap it can be if you try to make it be so. The thing you have to be aware of is how to make your coop no matter what you make it out of or who you buy it from completely predator proof. I can tell you how to do that easily.
I suggest you get pullets and not chicks with a cat around. My dogs don't bother the hens at all because I taught them not to. How your dogs will be is totally a function of what kind of relationship you have with your dogs.
My husband was really against the chickens at first. He said I could get three (the minimum number of chickens one should ever get) and if it didn't work I'd have to get rid of them, that they had to be "ugly" chickens that I wouldn't want to pet and make him eggs and not cost too much or be too much of a pain or too messy blah blah blah - all an experiment that he could pull the plug on at any moment. Years later he wanted more chickens, is out petting the chickens, delighting in them and laughing at them, made them a fancy waterer and built them a big coop and spends more time on chicken forums than I do!
There is nothing more fun than having walking clucking feathered clown ornaments that make you eggs in your yard. I spend way too much time on my chickens not because I have to (they pretty much take care of themselves) but because they are delightful.
I hope you are allowed to have them! If you are, I will tell you all I think you need to know and point you to some great websites.