The ghost feature (flying through "solid" virtual objects) is easy. Once he (Neo) finds the clipping plane controls, he'll be set.

The Ghosts appear to be normal Matrix Simulated Persons, with a damaged lockout on their clipping plane. The difference in appearance between modes is simply for effect, either on their part, or on the part of those who made them the way they are.

Neo flew faster to save the woman he loves, because she was dying.

The spoon tie-in may be more than just a memory trip. It was when Neo realized that "there is no spoon", that he came into being as "The One" ("Neo" is a prefix meaning "New" in Latin, and is also an anagram for "One". Considering the end of the movie, that's interesting.).
If Zion is really the second of two Matrices, then Neo remembering, and thinking about that event may have triggered his understanding of Matrix-Zion.

Agent Smith, from all evidence, is now no longer trapped by the constraints of Matrix-Earth. He is no longer required to obey the minimal structural, economical, or population-relational laws of the Agents.

Free from his bonds, Agent Smith is now Virus Smith, propagating himself at will in an attempt to corner and eradicate the Neo-Anomaly. This propagation is not limited to regular data systems, but can seemingly invade the living mind of a Matrix-Zion body.

Matrix Simulated Persona, Virus Smith, was able to transfer himself to a server (the human brain of the second messenger) independent of the Matrix network. This means that Virus Smith is not only controlling the body of the infected person, but that he can use that body's jacks to transfer himself to simulations external to the Matrix Network.

The Matrix Simulated Body of Real Persons appears to (if you recall the first film) be based on one's own reality-vision of themselves.
Thus, Virus Smith, in any vessel, is still represented in Matrix-type virtual space, by the Virus Smith Simulated Body.

I don't know if I'm correct, but it's interesting to hypothesize.