River whispers in his ear and tells him "his name" but in the Wedding of River Song the doctor didn't tell her his name, he said "look into my eye", did River do the same?
That's possible, and given that it's Moffat, maybe even likely. But it doesn't make sense to me. What I think:
The Doctor and River have to meet one more time, at the Fall of the Eleventh, when the Question is asked. That's when River learns his name. As for the Library, that depends on your definition of "living." I think she spends eternity in the Library computer. Which sounds like heaven.
yes! (in a way.) her body is dead but her mind is a live trapped in the computer system. She can still think and act on her own.
I think he told River his name some other time- possibly on one of her escapes from Stormcage? But River did die in the Library. Her consciousness is still existent though, so she's kind of alive, but not really.
if she does she'll die. everyone else was saved when they were fine. river's echo was all that was left. if it's released she'll fade
I don't think so - as everyone else said, she probably learnt it some other time (maybe at the Fall of the Eleventh?). And technically she's dead in the Library, but her mind has been 'saved'. I do hope she lives past the Library though!