general relativity - Definition of derivative operator on a manifold

I'm hoping to understand the motivation for certain parts of the definition of a derivative operator $\nabla$ on a manifold $M$. In Wald's General Relativity, two clauses of the definition are:

1. 
   

   Commutativity with contraction: For all tensors $\mathit{A} \in \mathscr{T}(k,l)$, $\nabla_{d}(\mathcal{A}^{a_1...c...a_k}_{b_1...c...b_l}) = \nabla_{d}\mathcal{A}^{a_1...c...a_k}_{b_1...c...b_l}$. (the parenthesis indicates contraction)

   
   



2. 
   

   Consistency with the notion of tangent vectors as directional derivatives on scalar fields: for all functions $f \in \mathscr{F}:M \rightarrow \mathbb{R}$ and all tangent vectors $t^a \in V_p$, it is required that $t(f)=t^a\nabla_af$

   
   

What is the point of having derivative operators commute with contraction (where we sum over the vectors and dual vectors for some (i,j) slot in the tensor)? What theorems are not possible to prove if this commutativity isn't stipulated? The so-called "Leibnitz rule" for derivative operators on manifolds corresponds to the simple product rule in elementary calculus; does this commutativity-with-contraction rule correspond to something simple as well?

My second question is about the notation of clause 2. We know that the $t^a$ are tangent vectors in the tangent space $V_p$ at point $p$ on the manifold. Then what is $t$ supposed to be?