It seems unlikely that River Song will appear, at least not regularly, in the 12th Doctor's tenure.
We have seen almost all of River's life on-screen, and other than her single appearance in Forest of the Dead / Silence in the Library, she was with the 11th Doctor:
- She was born in A Good Man Goes To War, and was kidnapped by The Silence
- She grows into a small child in their captivity, as seen in The Impossible Astronaut / Day of the Moon, where she escapes and regenerates into Mels.
- She grows up into a teen-aged Mels with Rory/Amy, then regenerates into River Song, as seen in Let's Kill Hitler.
- At the end of Closing Time we see her, now with a Ph. D in archaeology, kidnapped again by The Silence.
- She "kills" The Doctor in The Impossible Astronaut and/or Wedding of River Song, at which point she is imprisoned in Stormcage.
- She escapes from Stormcage numerous times, and eventually gets pardoned, each time progressively later in her life but earlier in The Doctor's life, as seen in The Pandorica Opens or Time of the Angels, for example.
- In the mini-episode "Last Night" we see The Doctor's last "date" with River, as the "third" (future-most) River tells The Doctor that he's taking her to the Singing Towers of Darillium.
- The next time she sees The Doctor is at The Library, where she dies.
Note that, when she meets The 10th Doctor, she immediately concludes that it's "very early days" for him because of his face. This is, as far as we know, the only time she meets this incarnation, so she probably doesn't recognize him directly. The most logical conclusion is that she knows she never meets any of his future incarnations, so any one besides the 11th must be earlier.
It's possible that she may appear in episodes of the 12th Doctor's lifetime, but so far we've seen no real indication of that. She's been notably absent from the shows since The Doctor lost Amy and Rory; her only appearance was as her deceased saved essence. To me, this implies that Moffet is trying to write her out of the series.
As far as the specific events you mentioned:
We almost have to assume at this point that she was told his name off-screen; perhaps they had a more formal wedding ceremony at some point, since their wedding in Wedding of River Song arguably never happened, and was rushed anyway.
River forgot almost everything that happened between her and The Doctor prior to "killing" him in The Impossible Astronaut, at which point he did know almost everything about her. But even in Let's Kill Hitler he knew a good amount about her future, enough to seem like he knew "everything".