Is a car rental customer responsible for maintenance and repair of there vehicles?

I would be very surprised if they were (at least in the United States). For my company at least (Enterprise), the customer is not expected to do any maintenance on the vehicle beyond the obvious (putting gas in it). Even for minor issues such as low tire pressure and washer fluid, our customers can either return to a rental branch, or visit one of our vendor gas stations or body shops for any needed service.

Any rental vehicle that requires repairs during the rental period is the responsibility of the rental company, as long as the damage is not the renter's fault. Even if the renter is liable for the cost, the actual repair of the vehicle is handled by the rental company and the customer will simply be assessed a fee.

If you do damage a rental car, even if it is your fault, DO NOT have repairs performed on it at your own expense, or perform any such repairs yourself. The rental agency has body shops for this, and there are legal reasons for not allowing customers to perform unauthorized maintenance on a rental car.

If you believe you have rented a car that was already damaged and the rental agency did not note this on your contract (cosmetic damage such as scratches or dents), you should contact the agency and have it recorded.

If you rented a vehicle that had something mechanically wrong with it, you should return it to the agency for replacement. If you are not comfortable driving the car, or it is unsafe to drive, leave it where it is and call your rental agency for instructions on how to proceed. Most will arrange to pick you up and/or deliver a replacement car.

Rental companies do not typically knowingly rent a defective car. Sometimes the problem is difficult to detect during the brief time it is inspected and driven on the lot, and such inspections are usually cursory. Routine maintenance will discover problems, but these service intervals are about 3 months. Sometimes even a new car will just have something randomly wrong with it. This is statistically expected given the high volume of cars in the fleet.