Most of the time the only rights a renter has are the rights the rental company sets. These rights differs from company to company; make sure to examine their policy carefully. Local laws can override this policy in some specific cases, but each area has their own.
Most rental companies offer expensive insurance plans to go with a rental, these can be waived if you prefer to use your own coverage, the only issue is you need to double check that your own coverage is applicable. For example American Express provides no extra coverage in Italy or New Zealand for car rentals. It is important to make sure the car is as covered as you can afford.
The major rental companies should provide road service where your car will be replaced immediately. If they cannot respond instantly, you should be able to get permission to have a local shop do the work and be reimbursed later. Make sure all of this is the case, because as mentioned before it varies from company to company.
