Answers
No problems will arise.
There is no need for additional documentation regarding the criminal records of the investor who applies for a permanent residency permit for property owners. The interested party must submit an application for an entry visa to the Greek consulate authority in his country of origin in order for entry in Greece and complete the necessary formalities of the procedure for property acquisition and the Golden Visa issuance. Since the investor has legally entered in Greece, there is no further documentation for criminal records needed. Furthermore, a certificate of a clean criminal record issued by the relevant authority of the home country is required when you apply for a long-term visa (not a tourist, type C one, which allows you to stay in the country for up to 90 days in a six-month period).
Criminal record is not needed to apply. So no impact.