Answers
You don't need a Schengen Visa to proceed with your Golden Visa application. You can realize the whole process through a Power of Attorney, signed either in front of the Greek Consulate near your permanent residence or in front of a local Notary and apostille/legalize it afterwards. Your authorized lawyer can upon this to submit your GV application and issue the temporary residence permit on your behalf, with which you can travel freely to Greece without any Schengen Visa.
The Golden Visa residence permit is the only one, for which the applicant can apply without being in Greece, on the condition that s/he has signed a Power of Attorney (PoA) for a Greek lawyer at the nearest Greek Consulate. As soon as the lawyer applies for the residence permit, a temporary residence permit is granted to the applicant, by which he/she can enter
Greece without holding a valid Schengen visa. In this context, the visa rejection will not affect the Golden Visa application at all.
No need in the beginning to get a visa. Just sign a POA at the local Greek embassy to a lawyer. He will buy property on your behalf and then you can enter the country legally.