Y'all have beaten me to all the great answers already!
dzheremi's link is helpful for the EO side of things, as we have the 15 autocephalous churches but many national churches that aren't self yet governing, but exist in the structure of other autocephalous churches. Then, the youngest of the autocephalous churches, the Orthodox Church in America, is not universally recognized as autocephalous. Those that disagree with the autocephaly see the OCA as part of the Russian Orthodox Church, since that is their "mother" church.
Really, to know if an EO church is autocephalous, you have to look at the diptychs (the list of bishops), and see who's under who. If the line doesn't end with one of the 15 autocephalous primates, they aren't canonical and we do not commune with them. Then, in turn, all of the other autocephalous churches commemorates the others when their primate is serving, thus declaring the communion between those churches. When a church is "struck from the diptychs", it means a schism has occurred.
As for EO/OO communion. It doesn't generally happen, but sometimes it is allowed (usually only when there isn't a local parish of the proper communion). As far as seeing each other as "Orthodox"...well, opinions will differ among the laity and even the clergy. Personally, I believe the OO to be completely Orthodox and would commune with them if I had the blessing to do so. That said, I don't have that blessing and so I do not. I hope it is a schism that will soon be healed.
When you speak of "subgroups" I think of schismatics groups, like the Greek Old Calendarists, who aren't in communion with us. Again, it is a simple better of referring to the diptychs. Their bishops aren't commemorated by the rest of the EO communion, and so they are not with us.