Custom theme for business directory

I have a project for a client which so far had decided to do custom from the beginning. But after reading about wordpress custom post types and custom taxonomies I’ve been trying to figure out if I would be better of using it instead.

What I want to know is if I have a custom type “Company” and create a custom taxonomy of “Companies” with terms like “Automotive”, “Education”, “Sports”… is it possible for each term to have a subterm so that finally I would end up with something like:

-Automotive
—Car Parts
—Car Painters
—Dealeships
—etc…
-Education
—Foreign languages
—Bookstores
—Tutors
-Sports
—Gyms
—Spas
—Tennis clubs
—etc…

And could I create an index page that displays all taxonomies and terms with links to each one, so for example when I click Gyms, it will show all the gyms (which each of them will be of a custom type “company”)

Can this be achieved with custom post types and taxonomies ? I know that many people are asking about the same questions I just haven’t been able to find the answer.

Note : I know there are ready-made templates and plugins for a business directory but I want to be able to make one myself so I can customize it any way I want. So without sounding bad, please don’t point me to plugins and themes.

Thank you for your time!