The Matrix

[Vesuveus]

1. His perception of what was happening.

2. The machines messed with Neo's body in the powerplant. The power he displayed was "from the source" and the Oracle explained that's what he felt when he killed those sentinels. He was endowed with the powers to serve his purpose.

3. I think one of the comics explained this. The bots took him for a deceased human and flushed him to be recycled. The crew of the Neb intercepted him

4. He was talking to either the Architect or Dues Ex Machina (the machine that jacked him in to fight Smith at the end of Revolutions). They monitored him with the phone tap which was interrupted. So he sent his message via payphone. btw, the intersection he is at in Downtown while making that call and flying off is the same intersection smith slammed him into the ground in Revolutions. If you press pause or slo-mo, you can see the phone booth.