Hey, I took a quick stab at your cage to illustrate what I meant by the first suggested approach (generally referred to as top down) to designing this thing. There is a base sketch that defines the foot print and a skeleton sketch that defines everything else. (there's a third sketch that resides in the door component that was necessary to create a parametric joint location for the door offset. only necessary b/c the offset you can enter in when defining joint locations aren't parametric. Hopefully AD will be fixing that soon.)
play around with the user parameters and see how things work. When you separate things out like this your BOM that is generated from the design will be correct, be cause it goes by components, and not separate bodies.
Next step is to insert a hinge from McMaster Carr. Then you can remove the rigid joint I used to place the doors and put a revolute at the hinge. Maybe a wide piano hinge across the top?
Next step after that is to further refine by going into each individual component and breaking them down in to their raw material shapes. each shape will then be a component also, but will reside inside it's master component. then your technical drawings will really rock. You'll be able to produce a drawing that shows the length and angles to cut for say the side of the door AND a drawing that show how the door goes together.
couple of notes on your workflow-
-your work flow is generally considered a bottoms up, or maybe sideways workflow, which is perfectly fine if that works for you. but always try to find ways to keep thing simpler, cleaner and more organized as you learn more.
-see above about components and bodies. You've got a joiner component with five bodies. That should just be five joiner components. (more organized)
-bodies are placed with the mirror command during the process of creating what will become a component. It shouldn't ever be necessary to mirror a component because they are placed with joints.(simpler,cleaner)
-might be fun to figure out how to get what you want without creating that extra plane (simpler, one less feature)
Also, on your sketches-
In fusion, it's a good idea to avoid mirror, offset, and pattern constraints when possible. This isn't a good design thing, it's a Fusion thing. There are still some weird buggy things that creep in with those types of constraints inside of sketches. They do work, until they don't...