Yes, every two-qubit state can be represented as a linear combination of tensor products of Pauli matrices (including the identity operator). The general formula for expressing a two-qubit state in terms of Pauli matrices is:
|ψ⟩ = (a I ⊗ I) + (b X ⊗ I) + (c Y ⊗ I) + (d Z ⊗ I) + (e I ⊗ X) + (f I ⊗ Y) + (g I ⊗ Z) + (h X ⊗ X) + (i X ⊗ Y) + (j X ⊗ Z) + (k Y ⊗ X) + (l Y ⊗ Y) + (m Y ⊗ Z) + (n Z ⊗ X) + (o Z ⊗ Y) + (p Z ⊗ Z),
where a, b, c, ..., p are complex coefficients that determine the state |ψ⟩, and I, X, Y, and Z represent the identity, Pauli-X, Pauli-Y, and Pauli-Z matrices, respectively.
In your example, |00⟩ can indeed be expressed as a linear combination of tensor products of Pauli matrices:
|00⟩ = 0.25(I ⊗ I + Z ⊗ I + I ⊗ Z + Z ⊗ Z).
The coefficients in the expression depend on the specific state you are representing and can be calculated based on the inner product between the state you want to represent and the basis states formed by tensor products of Pauli matrices.
It is worth noting that the set of Pauli matrices, along with the identity operator, forms a basis for the space of two-qubit states. This means that any two-qubit state can be expressed as a linear combination of tensor products of Pauli matrices (including the identity operator) using appropriate coefficients.