Answers
The private health insurance concluded in your country of origin must explicitly provide coverage for the whole length of your stay in Greece in order for it to be valid from a Golden Visa perspective. Otherwise, you can sign such an insurance policy here in Greece, choosing among the various contracts offered by the companies in Greece.
Yes, you are required to be insured and issue personal insurance policies for each family member.
The main applicant for a Greek Golden Visa is expected to obtain health insurance that is valid in Greece with some basic requirements (any aditional benefits would be your choice). As resident permit holders, you could purchase health insurance in Greece that is certainly valid in Greece.
You as an applicant, together with any minor children and/or husband and wife, need to make a private health insurance without any particular requirements and generally in low cost.
Among the requested documents to be submitted for the application of the Golden Visa is certification by an insurance agency for the cost of hospitalization and medical care. To certify that this condition is fulfilled, the following are accepted: insurance contracts which have been signed outside Greece, provided that they explicitly mention that they cover the interested party for the duration of their stay in Greece; insurance contracts which have been signed in Greece.
The requested minimum coverage that should be mentioned are described restrictively in the law. Of course, you have the option to arrange such an insurance contract in Greece and bear in mind that such contracts are cheap. The annual cost varies from 100 euros to 250 euros, depending on the age of the applicant, the insurance company, etc.
It is very easy to buy an insurance from Greece and much more convenient than proving that your insurance is valid. The cost is less than 200 euros and there is a list of insurance companies that issue specific programs for resident applicants.
Health insurance is bought in Greece. It only has basic coverage. It is necessary for the Golden Visa process.