The way zoning works is, cities are broken up into zones. These zones are labeled "residential" or "commercial" or "industrial" or "agricultural", or "residential 1" or "light industrial". Or something else, but the basic idea is the same.
Each zone has allowable uses to which your land may be put. There are also conditional uses. Allowable (or Permitted) means you can do this thing on your property without asking for permission. Conditional means it's Ok to do that thing, but you need to apply to the city to do so. If what you want to do is neither allowable or conditional, you have to apply to the city for a Variance.
In order to truly understand whether you can have chickens on your property, you first need to know what zone your house is in. Then you can look up the allowable uses for your zone. After that, I recommend searching "poultry", "chickens", "fowl" and "livestock" so you have a thorough understanding of how your city defines each. If you can classify chickens as pets, excellent. But it may be that there's some line in the definition of "livestock" that incorporates poultry.
In my city, the only verbiage related to chickens are in the "animal cruelty" section (no artificially-colored chicks), the "sanitation" section (no rats or smelly poop), and as they relate to kennels. My city defines a kennel as "more than 4 dogs, cats, or chickens". When I look in my zone, "kennel" is not listed as an allowable or a conditional use. Therefore, I may have up to 4 chickens on my property. We also have nuisance noise ordinances, which would apply to a rooster (or a barking dog, or a particularly enthusiastic construction crew).
Some cities have rules about where you can build chicken coops, or "accessory buildings", and how large these "accessory buildings" can be. Looks like yours are listed as "Area and Bulk Regulations". Make sure you know how far off your property line these must be, and don't exceed the maximum height. Or, build a "temporary" structure (tractor, hoop house, dog house).
Fencing may also be regulated under your code... make sure any fences you build don't exceed the maximum allowable height.
Ignoring fence regulations and requirements for accessory buildings (coop, shed, greenhouse... anything) gives your neighbors an excuse to complain to the city, so be careful.
Good luck! This is complicated, and sometimes city officials don't know what the code actually says. Your best defense is to carefully research, read, and try your best to interpret and understand the code as it is written. Good job finding it in the first place, you're on the right track!
