a) The convention, when differentiating a real valued function $v$ wrt to a column vector $p$ of dimension $(G \times 1)$, is that:
$$
\nabla _{p}v\left( p,w\right) \equiv \frac{\partial v}{\partial p}\left(
p,w\right) =\left(
\begin{array}{c}
\frac{\partial v}{\partial p_{1}}\left( p,w\right) \\
\vdots \\
\frac{\partial v}{\partial p_{G}}\left( p,w\right)
\end{array}%
\right).
$$
b) When expenditures $w$ are not constant, the column vector $p$ with respect to which we take the derivative occurs twice in $v(p, e(p,\overline{u}))$, and the chain rule has to be applied in order to obtain the total effect of the price change on the indirect utility: the first partial effect on utility is due to the price changes, and the second effect is through the adjustment in expenditures implied by the price change. Hence the expression:
\begin{eqnarray*}
&&\frac{\partial v}{\partial p}\left( p,e\left( p,u\right) \right) +\frac{%
\partial v}{\partial w}\left( p,e\left( p,u\right) \right) \frac{\partial e}{%
\partial p}\left( p,u\right) \\
&=&\left(
\begin{array}{c}
\frac{\partial v}{\partial p_{1}}\left( p,e(p,u)\right) \\
\vdots \\
\frac{\partial v}{\partial p_{G}}\left( p,e(p,u)\right)
\end{array}%
\right) +\frac{\partial v}{\partial w}\left( p,e(p,u)\right) \left(
\begin{array}{c}
\frac{\partial e}{\partial p_{1}}\left( p,u\right) \\
\vdots \\
\frac{\partial e}{\partial p_{G}}\left( p,u\right)
\end{array}%
\right).
\end{eqnarray*}