1. When did the Doctor tell River his name?
2. Who gave Clara the Doctor's phone number?
3. If Clara is meant to have been saving the doctor whenever his life was in danger why didn't she take River's place in the library?

Because she was just putting things back as they originally were not changing things

But I thought 'how things originally were' was how things were with her in the frame?
- I'm sure there are plenty of other people who have died to save the doctor, but River is the one that I know of. If she was always there in the background why did she let other people die when she was the one who was meant to be saving him? or was it that she would save him if no one else was prepared to

She was there to prevent whatever the GI did, not get him out of any kind of danger that he'd have escaped anyway.

In the original history River ended up in the library so Clara didn't mess with that or anything else that originally happened even if times the Doctor's friends died because changing it could have made things worse over all and changed The Doctors past

Time-wimey.

The Doctor saved all those planets in the past, until the GI stepped in to interfere. Which means history was changed. Clara stepped in to simply undo that, working in the background to ensure the Doctor didn't fail, or to ensure the GI didn't get to stop him. We don't actually know what Clara did in all those millions of timelines, but I suspect it was stuff like with Hartnell. The GI nudged him to take a different Tardis, Clara nudged him back.

We also know, because Clara said so many times, that the Doctor often didn't hear or acknowledge her presence. So she was very much in the background, just helping things turn out right again.