(1) is Théorème 7.4.18 of BT1: Bruhat and Tits - Groupes réductifs sur un corps local. Because of (1), to show (2) (as you define it, i.e., transitivity of the action of $G$ on apartment–chamber pairs) it suffices to show that the stabiliser of a chamber acts transitively on the apartments containing that chamber. This is Corollaire 7.4.9(i) of [BT1].

(1) is Théorème 7.4.18 of [BT1]: Bruhat and Tits - Groupes réductifs sur un corps local.

Now let $(A_1, C_1)$ and $(A_2, C_2)$ be two apartment–chamber pairs. Upon replacing $(A_1, C_1)$ and $(A_2, C_2)$ by $G$-conjugates, we may, and do, assume that $A_1$ and $A_2$ both equal $A$. Now the entire computation is happening with respect to a fixed affine root system, and has nothing to do with the full valuation of root datum, so we go back to the very beginning. (1.3.3) of [BT1] shows (or, rather, quotes from Bourbaki the fact) that there is some element of $W$ that conjugates $C_1$ to $C_2$. This element of $W$ may be lifted to an element of $N \subseteq G$. Per your definition, this is what is required in (2).