My tenant(s) have not paid their rent, do I have to fix broken things in the rental unit?

Yes, you should still make attempts to make repairs. Even if your tenant is not paying the rent, they can make a claim that you violated the warranty of habitability if they complain to you about a repair that is needed and you fail to fix it. Your obligation to make repairs is not tied to the tenants' obligation to pay rent. In other words, a tenant not paying their rent does not cancel your landlord obligations.