@Francis A.
It depends if you want to rent to section 8 or not. If you don't, you can tell prospective tenants, or put in your ad that you do not accept section 8. If you are willing, then I don't think you need to be pre-approved. Section 8 does an inspection, which is pretty basic - no double-cylinder locks, no obvious vermin, etc. If you pass, then you enter into a section 8 contract.
As for saying no pets, you can say that you don't accept pets, but as you probably already know, without citing the sections of the federal (ADA) , California state (FEHA) , and municipal laws (I am sure there are some LAHCID regs on this), I will just say that you cannot legally discriminate against disabled tenants who have service animals by claiming it violates your a no pet policy.
Service animals are not pets and if you refuse to rent to a potential tenant based on their disability and their service animal you are opening yourself up to substantial liability. Even if you discourage or "chill" a potential tenant by saying "no pets" when they ask about a service animal, that could be actionable. Civil rights lawyers or fair housing groups will often take cases in borderline situations like this assuming that they will be able to find more evidence of bad behavior as the case goes on.
Edit: craigslist probably flagged no pets as they have a dog/cat button on the post that you check or not.