I have three questions regarding height pairings:


1. In [Serre, Lectures on the Mordell-Weil theorem], p. 85 f., it is stated that the following function is a local height function: 
"Let $V/R$ be a scheme over a discrete valuation ring, $D$ a divisor on $V$ and $x \in V(R)$. Assume $x \not\in \mathrm{supp}(D)$. Define $m(x,D)$ as follows:
a) If $x$ does not intersect $D$, $m(x,D) = 0$.
b) If $x$ intersects $D$ in a point $x_0$ of the special fibre, then $m(x,D)$ is the intersection multiplicity of $x$ and $D$ and $x_0$.
Then $x \mapsto m(x,D)$ is a local height associated to $D$." Does anyone have a reference for this?

2. Why is for a curve $X$ the Néron-Tate canonical height $\hat{h}_A(x,\mathcal{L})$ equal to $\mathrm{deg}_X((x,\mathcal{L})^*\mathcal{P}_A$ with the Poincaré bundle $\mathcal{P}_A$?

3. We have (functoriality of the height) $\hat{h}_{f^*(\mathcal{M})} = \hat{h}_{\mathcal{M}} \circ f$. Can one prove this also with the other definition of the height in 2.?