Relationship between Multiplicative Ergodic Theorems

The equivalence of "sublinear shadowing" property and the Oseledec theorem (or, rather, of what is called "Lyapunov regularity") is due to Kaimanovich MR0947327 (89m:22006), and is explained there in detail (actually, this is a purely geometric property which has nothing to do with random products). By the way, your formulation of the Oseledec theorem is incomplete: one should add that (4) the sum of the Lyapunov exponents (taken with their multiplicities) should be equal to the exponent of the determinant (the expectation of $\log\det A(\omega)$).

It seems to me that the OP is asking specifically what the metric on the symmetric space is, and why it is non-positively curved. These facts are discussed in Bhatia's paper (2003), where he refers to Lang's 1999 differential geometry book and Ballmann/Gromov/Schroeder for proofs (Bhatia's paper is relevant to the question as well).