Driving around Bali?

Hi,

I'm planning to visit Bali for the first time in the beginning of June. I'm a Scandinavian who's lived in Bangkok for the past 10 years and visited all the neighbouring countries except Indonesia. I've never had interest in the busy beaches of Bali, because there are enough of those here in Thailand, but what I would really like to do is rent a car for the whole week and explore the island. I would be interested in the less touristic spots such as quiet and scenic beaches, maybe the Ubud region and any possible scenic driving routes.

So I would have a few specific questions before I book my flight tickets:

- How is the traffic and driving culture compared to Thailand? I've been driving a car in Bangkok for some 4 years now without any issues. Shouldn't be a problem to drive in Bali or? How much should I expect to pay for renting a car for a week?

- Is the beginning of June a good time to visit? I read that it's kind of low season so would there be a bit less crowds and better hotel prices? And the weather, is it a bit more dry in June?

- Any suggestions for a few beautiful beaches that are not too touristic? If I have a car it wouldn't matter if they are harder to get to either.

Thanks!