What things can you unlock or customize for your apartment?

You obviously have the gun wall. There is a seperate question specifically for the gun wall so I won't cover this in depth here but this wall is populated by putting specific iconic weapons into your stash.

You can get a cat - the cat is available by leaving your apartment and immediately going left to the end of the corridor. Just around the corner there is a bowl next to a note saying 'feed the cat'. If you put cat food in the bowl and then come back the next day, there will be a cat that you can take back to your apartment.

If you complete the side quest Fool on the Hill, Misty will gift you a dreamcatcher that you then hang above your bed. You don't have to hang it, it will be hung for you automatically.

After completing the Riders on the Storm side job, you will unlock another small job called I'll Fly Away, which is Scorpion's funeral. At the end of his send off Mitch will give you a Scorpion figure, which you can then put on the table in your apartment.

For completing the Peralez side mission Dream On, you get a Vote Peralez campaign poster, which is automatically put up on the wall next to the front door to your apartment. This seems to be the case even if you do not finish the missions on good terms.

Likewise if you complete the Lizzy Wizzy side mission Violence you get a Lizzy Wizzy poster to go next to your Peralez poster. This seems to be the case even if you don't stop her from going psycho.

Depending on your player back story, you will either have a Night City model, an Arasaka paperweight or a car model on your desk, after completing some of the first missions made available to you in the game.

After completing Following the River, you will get a copy of Big Trouble in Heywood (the augmented reality game you played with Joss's kids).

If you complete Judy's side quest Pyramid Song, you also unlock some scuba gear that is stored in your apartment.