Since he does not live in the country, should I sue him or the insurance company or the rental car company for the accident?

You should consult with an experienced personal injury attorney. While you can sue the rental car company without suing the driver, it may be a poor tactical decision to do so; suing a resident of another country is not difficult, but serving them with a summons and complaint is not something you even want to try. When you call the lawyer, ask him or her if they have ever sued a non-resident alien, because the rules which apply to that are a bit out of the ordinary.