Let graph $P$ be a regular 2016-gon.
(1) If a diagonal of graph $P$ divides $P$ into two parts (the edges of $P$ are not considered as diagonals), and each part contains an even number of edges of $P$, then this diagonal is called a good diagonal. A convex $m$-gon (with $4 < m \leqslant 1008$) whose sides are good diagonals and whose vertices are vertices of $P$ is called a good convex $m$-gon. Calculate the number of good convex $m$-gons;
(2) If a diagonal of graph $P$ divides $P$ into two parts (the edges of $P$ are not considered as diagonals), and each part contains an odd number of edges of $P$, then this diagonal is called a bad diagonal. A convex $m$-gon (with $4 < m \leqslant 1008$) whose sides are bad diagonals and whose vertices are vertices of $P$ is called a bad convex $m$-gon. Calculate the number of bad convex $m$-gons.
(1) Let the vertices of graph $P$ be $A_{1}, A_{2}, \cdots, A_{2016}$, and the vertices of a good convex $m$-gon be $A_{i_{1}}, A_{i_{2}}, \cdots, A_{i_{m}}$. Clearly, the $m$ vertices of a good convex $m$-gon are all points in $\left\{A_{2}, A_{4}, \cdots, A_{2016}\right\}$ or $\left\{A_{1}, A_{3}, \cdots, A_{2015}\right\}$. Therefore, the number of good convex $m$-gons is $2 \mathrm{C}_{1008}^{m}$.
(2) From the problem, we know that $m$ must be even. Let the vertices of graph $P$ be $A_{1}, A_{2}, \cdots, A_{2016}$, and the vertices of a bad convex $m$-gon be $A_{i_{1}}, A_{i_{2}}, \cdots, A_{i_{m}}$.

Define the length of each side of a bad convex $m$-gon as the number of edges of graph $P$ contained in the arc that does not contain any other vertices of the bad convex $m$-gon, not including the vertices of the bad convex $m$-gon. Let the lengths of the $m$ sides of a bad convex $m$-gon be the set
$$
\left\{\left(x_{1}, x_{2}, \cdots, x_{m}\right) \mid A_{i+1}-A_{i}=x_{i}\left(1 \leqslant i \leqslant m, A_{m+1}=A_{1}\right)\right\}
$$

Clearly, $x_{k}(1 \leqslant k \leqslant m)$ is a positive odd number greater than or equal to 3, and
$$
\left.x_{1}+x_{2}+\cdots+x_{m}=2016 \text { ( } m \text { is even }\right) \text {. }
$$

For each bad convex $m$-gon, the lengths of its $m$ sides are solutions to the indeterminate equation (1).
On the other hand, for $i_{1}=1,2, \cdots, 2016$, selecting points in a counterclockwise direction, each solution to the indeterminate equation (1) corresponds to a bad convex $m$-gon, but the bad convex $m$-gons corresponding to all solutions to the indeterminate equation (1) will have $m$ bad convex $m$-gons that are the same.

In fact, if a solution to the indeterminate equation (1) is $\left(a_{1}, a_{2}, \cdots, a_{m}\right)$, and this solution corresponds to a bad convex $m$-gon $A_{j_{1}} A_{j_{2}} \cdots A_{j_{m}}$, it is easy to see that the bad convex $m$-gon $A_{j_{2}} \cdots A_{j_{m}} A_{j_{1}}$ corresponds to a solution to the indeterminate equation (1) of $\left(a_{2}, \cdots, a_{m}, a_{1}\right)$.
Therefore, there will always be $m$ bad convex $m$-gons that are the same.
The number of solutions to the indeterminate equation (1) is $\mathrm{C}_{\frac{2016-m}{m-1}-1}^{m-1}$, so the number of bad convex $m$-gons sought is $\frac{2016 \mathrm{C}_{\frac{2016-m}{m-1}-1}^{2}}{m}$.