(1) What is the Knowledge Grid? What is the Knowledge Grid Methodology?

The semantics of the "Grid" within the "Knowledge Grid" is different from the "Grid" within the "Grid Computing".   The Knowledge Grid concerns effective knowledge sharing over various networks and the semantic basis for knowledge sharing.
The following paper would help you to find the answers:
H. Zhuge, The Knowledge Grid and Its Methodology, 1st International Conference on Semantics, Knowledge and Grid, Nov 27-29, 2005, Beijing, China.

(2) What is the relationship between P2P and the Knowledge Grid?

A semantic overlay built on P2P network supports a scalable Knowledge Grid.
The following paper introduces a general approach to construct a semantic overlay on P2P network.
H. Zhuge, X. Sun, et al,  A Scalable P2P Platform for the Knowledge Grid, IEEE Transactions on Knowledge and Data Engineering, 17(12) (2005) 1721-1736 (PPT).

(3) What is the fundamental scientific issues of the Knowledge Grid ?

The Knowledge Grid has three fundamental issues: 1) effective organization model for distributed resources, 2) semantic networking model, and 3) dynamic clustering of distributed resources.