Yes you can install from the disc (it'll have quite a few updates to do though!), but you shouldn't need your code at all.

When it asks if you have an SE account or not, just say yes and give it the login details of your existing account. It should then check that, see there's an active subscription and carry on without asking you for anything else.

Unfortunately your hotbar settings were stored in My Documents\My games\FINAL FANTASY XIV - A Realm Reborn so if you haven't backed that folder up before your reinstall, you'll need to redo all the hotbars.