Is it incumbent upon my cousin and me to jointly bear the cost of repairing the car, half of which I rented from him, after it overturned due to a wheel malfunction while I was driving it at a reasonable speed, or am I solely responsible for this cost?

If there was no negligence or transgression on the part of the tenant, then he is not obliged to pay for the car's repair costs. This is because his possession is one of trust (yad amanah), and he is only liable in cases of negligence. The leased item is a trust in his possession; if it is damaged without his negligence, he is not liable for it.