### Part (a)

1. **Define the function $d(X)$ for $X \in S(T)$**:
- Choose an arbitrary point $T_0 \in T$.
- Define $d(X)$ as the distance from $X$ to $T_0$.

2. **Boundedness of $S(T)$**:
- Notice that $S(T)$ is bounded by a circle of radius $2012 + 2013 = 4025$ centered at $T_0$.
- This is because any point $Q \in S(T)$ satisfies $|QP| \le 2013$ for some $P \in T$, and $|P_1P_2| \le 2012$ for any $P_1, P_2 \in T$.

3. **Closedness of $S(T)$**:
- $S(T)$ is closed. For any convergent sequence of points in $S(T)$ whose limit has a distance strictly more than $2013$ from all points in $T$, we can find some point far enough along in the sequence that it too has a distance strictly more than $2013$ from all points in $T$, contradicting its membership in $S(T)$.

4. **Compactness of $S(T)$**:
- Since $S(T)$ is both bounded and closed, it is a compact set.

5. **Existence of a maximum for $d$**:
- The continuous function $d$ on $S(T)$ attains a maximum at some point $S_0$.

6. **Constructing a line in $L$**:
- The line through $S_0$ perpendicular to $T_0S_0$ will intersect $S(T)$ at exactly one point, $S_0$.
- Therefore, $L$ is nonempty.

### Part (b)

1. Definition of a bad line:
- A line $\ell_0$ passing through $O$ is bad if there does not exist a line $\ell \in L$ parallel to (or coinciding with) $\ell_0$.

2. **Sweeping lines parallel to $\ell_0$**:
- If $\ell_0$ is bad, then sweeping lines parallel to $\ell_0$ across the plane means each one that intersects $S(T)$ intersects it multiple times.
- This implies it intersects the convex envelope of $S(T)$ multiple times.

3. Convex envelope and line segments:
- The convex envelope of $S(T)$ must have a finite number of line segments of positive length.
- Suppose, for the sake of contradiction, there were an uncountably infinite number of (positive length) line segments.

4. Lemma: Sum of an uncountably infinite set of strictly positive numbers is infinite:
- Argue by contradiction and assume it were finite and equal to some $M$.
- For each $n \in \mathbb{N}$, the set $\{x \in S | x > 1/n \}$ is finite (in particular, has size at most $Mn$).
- The union of these sets is $S$, but the countable union of finite sets is countable, contradicting the uncountability of $S$.

5. Contradiction:
- It follows that the convex envelope of $S(T)$ must have infinite perimeter, but finite area (from the boundedness of $S(T)$).
- Consider the set $S'(T)$ of points which are within some $\epsilon > 0$ of some point in $S(T)$; clearly this figure has bounded area as well.
- By convexity, $[S'(T)] - [S(T)] \ge P\epsilon$ where $P$ is the perimeter of the convex envelope of $S(T)$.
- This contradicts the fact that $S(T)$ has infinite perimeter.

6. Countability of bad lines:
- Therefore, the set of bad lines is countable.
- We can assign a line $\ell(i)$ to each positive integer $i$ so that for every bad line $\ell_0$ passing through $O$, there exists a positive integer $n$ with $\ell(n) = \ell_0$.

$\blacksquare$