"I have read that the Calvinists AND Melanchthon reached out to the Patriarch, but as soon as he realized what they were about, he rebuffed them."
The Lutherans certainly did, in a most arrogant fashion too after the EP refused to buy in on some of their ideas which he called innovations, I might add. I've never heard that Calvin did. The text of the letters used to be posted on the internet but the site is down.
I vaguely remember reading that the Calvinists made some inroads in eastern Europe among the Orthodox. They initially had much support in Poland, which was, of course much larger than today. Incidentally, I have been surpirsed to learn that Calvin was very pragmatic. He would have allowed some form of the mass to continue and only went along with the radical scheme. He was also willing to live with bishops. So in general terms, he and English Archbishop Cranmer were very close together.