You never know what might happen while in a foreign country so we highly recommend you to have good travel insurance. Sure it costs a bit extra, but it can take a load off the mind and end up being a totally worth it in case something sudden could happen.
Take a look at what type of insurance is provided with your credit card, alternatively buy special travel insurance from an insurance provider.
As for the car rental, you can opt to buy extra insurance at the time of collecting the car. It may be wise to buy extra insurance such as windshield protection, as a cracked windshield from a stone being thrown off another car is among the most common incidents on Icelandic roads.
Other extra car insurances would be for example sand and ash protection, or additional insurance to lower the self-risk in case of an incident.