Re spoiler... no he didn't. They *both* lied. What he told her was not his real name. TWoRS actually shows you what he said.
I've come to the conclusion that River died in the Forest of the Dead...her future version in the past.
Yes, that's the case - it's the first time he meets her, and the last time she sees him. But the point where he tells her his name, the one time he could - we haven't seen that yet. We haven't seen their first meeting (from her POV) at the towers of Darillium, where he gives her his screwdriver.
I'm trying to understand why she came back to the past just to save The Doctor in the end because she knew he would die and cease to exist in the future somehow.
Because she didn't go back to the past. Her timeline is continuous - forwards. After she travels to the 51st Century and becomes a doctor, she goes to Stormcage for the 'murder' of the Doctor. She eventually receives a pardon, because by the time of FotD, she's no longer working 'to earn a pardon', she's essentially free, but doing this job as a job. From her timeline POV, Season 6 (except the epilogue of TWoRS) happens first (in the correct order, once we get past TIA and DotM), then the events in series 5 (Pandorica first, then the Time of Angels), then FotD.
The point where he shows up at the towers of Darillium, on her 'doorstep, with a new suit, and knew everything about me', that's when she first meets him from her POV, and that hasn't happened yet. (From the Doctor's POV, it must happen at any point from here on in, and from our POV, it'll be the last time we see her.)