Answers
In no case does the resident permit for property owners 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. So not even as freelancer you could exercise your business. The only available options are the above mentioned roles.
You can still create a company in Greece and be the shareholder.
It is quite difficult to answer this question. It really depends on which kind of activity you exercise and if income is produced in Greece. As a general rule, for activity exercised in Greece and income produced in Greece, you need to declare it to the fiscal authorities and you need to have the license. Now it all depends on how this will be handled, where money will be received and for which legal reason.
Greek Golden Visa holders cannot have a working permit in Greece, not even as a freelancer. After seven years of residency in Greece, a Greek Golden Visa holder could apply for Greek citizenship and then be eligible for full privileges.