Categories for CPTs?

What’s the proper way to create and manage categories for a CPT?

I’m not sure what to expect but here’s what I have:
-I’ve created a CPT named DOGS
-It has both Has Archive and Hierarchical set to true
-It has both Categories and Tags checked for Built-in Taxonomies

Once saved, I see the CPT in the Admin menu, along with its sub-menus:
-Add New
-Categories
-Tags

I can post to DOGS and the posts display just fine on the front end via:
mysite/dogs/my-post

However, I’m not sure how to manage the categories for DOGS.
When I click on the Categories sub-item for the DOGS CPT, it seems to take me to the main WP Admin for Posts>Categories even though the menu shows DOGS>Categories.

Then, if I do create a category there (e.g. Poodles), it shows up on its own (not under the parent DOGS because DOGS is not a category).

So, I created a DOGS category and attached DOGS posts to it. I get that archive to display in two areas:
mysite/dogs/
mysite/dogs/post-here

Though I’m thinking that mysite/dogs is really showing me the main CPT archive and mysite/categories/dogs/ is showing me the archive for the DOGS category.

After creating a Poodle subcategory of DOGS I also get a post in that subcategory to display in two areas:
mysite/dogs/post-here
mysite/categories/dogs/poodles/post-here

However, all I want is the following:
mysite/dogs/ –showing all DOGS posts
mysite/dogs/poodles/ –showing all DOGS/Poodles posts
mysite/dogs/poodles/post-here –showing a specific DOGS/Poodles post

. . . exactly how a regular WP post hierarchy would work.

Is this possible and how?

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