Insurance for expats in the Netherlands

As a rule, yes. Anyone living in the Netherlands or paying wage tax here is required to insure and must take out health insurance within the statutory period after arrival; cover applies retrospectively from the moment the obligation arose. There are exceptions, for example for a short posting with an A1 certificate. Have your situation assessed before you assume you are exempt.

Not once the car has a Dutch registration. Article 2 of the WAM requires the registered keeper to hold at least third-party liability cover with an insurer permitted to operate here. If the foreign registration stays in place, that country's rules apply plus a valid green card, but on permanent settlement that is usually not a sustainable route. Check this before you start driving.

Sometimes. Dutch insurers can take over foreign no-claims years on the basis of an original statement from your previous insurer, but they are not obliged to and apply different maximums. Request that statement before you leave; obtaining it afterwards often takes a long time. Without it you start at the lowest step and pay more premium.

No, not for private individuals. You are, however, liable yourself under Article 6:162 of the Dutch Civil Code for loss you unlawfully cause to another person, and that loss can run high. Landlords and schools therefore often assume that you have such cover. The insurance usually applies to all family members living at the same address.