Let $M$ be a riemannian manifold. For each $k\geq 0$, let $ \bigwedge^{k}(M, \mathbb R)=\bigsqcup\limits_{p\in M}\ \bigwedge^k(T^*_pM)$ denote the smooth bundle of ($k$-multilinear) alternating forms on $M.$
If $\alpha\in\bigwedge^{k-1}(M, \mathbb R), \beta\in\bigwedge^{k}(M, \mathbb R)$ and $v$ is a tangent vector, I want to prove that $$(1)\qquad \qquad\langle\!\langle v^\flat \wedge \alpha, \beta\rangle\!\rangle = \langle\!\langle \alpha,v\lrcorner\beta\rangle\! \rangle,$$ where $v^\flat(w) := \langle v,w\rangle$ and $\langle\!\langle \,, \rangle\! \rangle$ denotes the standard inner product on $\bigwedge^{k}(M, \mathbb R)$.
That is, $\lrcorner$ and $\wedge$ are actually adjoint operators with respect to $\langle\!\langle \,, \rangle\! \rangle$. I have seen this statement at several isolated places, but I haven't seen a proof of this (maybe it's not hard at all and I'm just not getting it, so it would be cool to know why this formula actually holds).
Also, if $\gamma\in\bigwedge^{k}(M, \mathbb R)$ and $v, w$ are tangent vectors, I want to see why the following Clifford relation holds $$(2)\qquad\qquad v^\flat \wedge (w\lrcorner\gamma)+w\lrcorner(v^\flat\wedge \gamma)=\langle v, w \rangle\gamma.$$ Finally, Willie Wong states here that there exists a constant $C$ (which depends a bit on the conventions and also on the dimension and the degree of the forms) such that $$(3)\qquad\qquad v\lrcorner\gamma = C \star(v^\flat\wedge \star\gamma), $$ where $\star$ is the Hodge star operator. How can I prove this formula?(which in turn also must determine the exact value of $C$)
I apologize if these formulas are easy to prove (I collected some "trivialities" usually left as an exercise to the reader), but I find them very pretty and somewhat cryptic, I'm not really sure on how should I proceed. Also, it seems that nobody actually wants to write out all the (possibly gory) details (or at least to explain them), so any help or reference is highly appreciated.