The non-null assertion that lied

A BFS over string-keyed graphs compiles clean under strict: true. At runtime, building the adjacency map throws TypeError: Cannot read properties of undefined (reading 'push') on the very first edge. A teammate "fixed" it by pre-seeding the map, and now every distance except the source comes out as NaN. Both failures trace back to the same character. Find it.

1function buildAdj(edges: [string, string][]): Map<string, string[]> {
2  const adj = new Map<string, string[]>();
3  for (const [a, b] of edges) {
4    adj.get(a)!.push(b);        // TypeError on the first edge
5    adj.get(b)!.push(a);
6  }
7  return adj;
8}
9 
10function bfsDist(adj: Map<string, string[]>, src: string): Map<string, number> {
11  const dist = new Map<string, number>();
12  dist.set(src, 0);
13  const queue: string[] = [src];
14  let head = 0;
15  while (head < queue.length) {
16    const u = queue[head++];
17    for (const v of adj.get(u)!) {
18      if (!dist.has(v)) {
19        dist.set(v, dist.get(v)! + 1);   // meant dist.get(u) — v is not in the map yet
20        queue.push(v);
21      }
22    }
23  }
24  return dist;
25}
26 
27// after seeding adj by hand:
28// bfsDist(adj, 'a') → Map { 'a' → 0, 'b' → NaN, 'c' → NaN }

1. What is the return type of Map.get() and what does the ! operator tell the compiler? What code does ! emit at runtime?
2. Explain the crash in buildAdj and why tsc accepted the line.
3. In bfsDist, dist.get(v)! + 1 produces NaN instead of throwing. Walk through the JavaScript semantics that make that happen, and find the typo the ! concealed.
4. Rewrite both functions without any !. What idioms replace it (??, get-or-create, an early undefined check)?
5. When is ! legitimate? Give one example from this exercise where an assertion would actually be justified.