God bless you.
I will form a new thread for you when I return home. Thank you for your question.
I would also be interested in the history of splits and etc. Thanks in advance!
One of the commonly pointed-out problems with the GOC is that there are now several claimants to the title "Church of the Genuine Orthodox Christians of Greece." The criticism goes that if the GOC is the real Orthodox Church, why is it internally divided; this is proof it is schismatic, etc.
I can only agree that it is a problem, although I think there are two considerations. The first is that this is not the first time that a group resisting a heresy fractured in the face of the immense difficulty of the situation (I think of the aftermath of the Council of Nicaea for instance). The situation during the majority of the history of the GOC was one of persecution (churches being seized, etc) and that played a part in it. Eventually in the Nicene case, the disparate parties rejoined and vanquished the Arians together. The other consideration is that our ordinations are given more recognitions than our depositions (!) so for instance, if we ordain someone a bishop, everyone calls him bishop, even if the New Calendar folks don't recognize him sacramentally, they still will accord him the courtesy of using the title and treat him as "the Old Calendar bishop of x" but then if that bishop is deposed by the Synod, they don't seem to accept that we deposed him. When this person goes off and starts his own Church, suddenly we hear that "Look at those Old Calendarists, they are all split up." No one takes the time to do some simple stats--there is one Synod that makes up the vast majority of the GOC, has the majority of the parishes, monasteries, bishops, clergy, people, and historical places, documents, etc.--the Synod of Archbishop Chrysostomos II (Kiousis). The other split offs together compose a minority and are in ever-shifting alliances and combinations. I think of the split offs that arose in America in the 1920's all calling themselves the American Orthodox Church--only the Church of Metropolitan Platon, which eventually fixed its relations with the other churches--was ever the "real" Metropolia. But having said that I do understand that the situation is disconcerting.
HOCNA is a unique situation in that they did not come from the GOC of Greece but rather were with ROCOR. When they split with ROCOR instead of joining the main Synod of the GOC, they joined with a deposed former Archbishop of the GOC who still had a small following. Now that almost all those players are deceased, though, there doesn't seem to be any reason to be separate and I believe something could be worked out.
The other Synod you might see on the internet is that of Metropolitan Cyprian (the Synod in Resistance) which consists of a group of archimandrites who were consecrated in irregular circumstances and attempted to purify the Synod of perceived problems. The majority of these new bishops were reconciled but Metropolitan Cyprian and one other bishop stayed separate because they did not believe the problems that precipitated the original action were resolved. They are the Synod which has a self-described moderate position towards the New Calendar, and were in communion with ROCOR for some years until the latter decided to rejoin the Moscow Patriarchate.
The above is a gross simplification due to lack of time. One can find the various positions and documents floating about the internet.