I agree that ripping 2x4's on a table saw is kind of a pain and dangerous if you are not used to it.
I can't get 2x2's here and have to rip them. I would only used lumber that small something like the tractor I am working on, with the coop section only being 3'x4'.
I know what you mean about feeling that working with smaller dimensional lumber would be easier, because I felt that way myself when I started building. I am also a woman, and when I started building chicken coops I didn't even know how to use a drill.
I built things with 1x3's and 1x2's at first, but I have since learned that it just doesn't make much sense to do that unless what you are building is very small. And using 2x4's shouldn't be more intimidating in practice. If you have a good little miter saw (mine is a small Ryobi that I got for about $100 from Home Depot) than cutting a 2x4 is no different than cutting a 2x2. and what you build will have a better chance of holding up to time and weather. That was my first, hardest lesson - build something poorly with small lumber, and you will soon have a piece of unsightly unusable junk in your yard that needs dismantling/rebuilding. But take your time, use the right stuff, and you will have a great useful coop for years to come.