Can logged-in customers use previously filled addresses in the checkout?

Yes, this is built-in to Commerce. When logged in users visit the checkout after placing an order previously, it will show their previously used shipping and/or billing address to reuse.

Additionally, the core-provided User Profile Address module will prefill the "new address" form with the information from the user profile. That can also update the address on their profile. This module is mostly useful if other parts of your site rely on the address on the profile being accurate/filled.