Answers
Regarding taxes, for a business based in Greece you will need to pay taxation applicable to Greece. Please advise also your accountant if that kind of activity serves you better, as your clientele is based abroad.
The resident permit for property owners does not provide access to any type of employment. Employment, according to paragraph 6, article 20, of Law 4251/2014, does not include the exercise of economic activity in the capacity of a shareholder or a chief executive officer for an already existing company. Following the above legal provisions you are not allowed to run your own business as you describe. If you are interested to run your home business in Greece you have to contact an experienced accounting or consulting firm to guide you accordingly and following Greek taxation legislation. Also take into consideration the provisions for tax residency. For income generated in Greece you have to pay taxes in Greece.
You cannot work from anywhere. But you can be the shareholder of a company.
The resident permit of the Greek Golden Visa allows the applicant to live in Greece but not to work. However, the applicant can set up a business in Greece. The income which is produced in Greece is to be declared and to pay taxes on it. The amount of money which is income from abroad (not from Greece) is to be declared, if there is a purchase to be justified in Greece, not to pay taxes.