(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).