First, don't worry about the asbestos insulation. So long as asbestos is not disturbed, it is safe. Since you're talking about insulation, I assume that asbestos is not floating around in the are, which is UNsafe.
You definitely can take you landlord to court for damages for cockroach infestation, ESPECIALLY if you have reports from the Health Department regarding cockroach infestation. Take the reports, and a lot of nasty photos, to court with you. If he doesn't fix the problem, you can sue him EVERY MONTH thereafter for the same thing.
(P.S. Why isn't anything being done? Call the Health Department and try to have this conversation with someone in charge.)
Call a landlord tenant lawyer and have them review your documentation and have them advise you as to how to proceed. The landlord does have a duty to rent a premises that is safe and habitable. Did her correct the cockroach problem? Did you notify him? Do you have documentation here. There are a lot of unanswered questions.
I'm sorry that you're going through this. Without knowing all of the facts, a responsible lawyer won't advise you.
This question is difficult to answer without more facts. It is true that the landlord has the duty to ensure habitability pursuant to Civil Code sections 1941 and 1941.1. However, whether you would win damages in a small claims court lawsuit would depend upon the extent of the cockroach infestation and how unreasonable the landlord has been in correcting the problem.
As for asbestos, many older buildings have asbestos in its building materials such as insulation. Unless the landlord did something to disturb the insulation, however, the landlord does not likely owe the tenant any duty to disclose or correct.
A landlord does have a duty to disclose certain topics, as described in:
Frank W. Chen has been licensed to practice law in California since 1988. The information presented here is general in nature and is not intended, nor should be construed, as legal advice for a particular case. This posting does not create any attorney-client relationship with the author. For specific advice about your particular situation, consult with your own attorney.