Fix API build: shared package emits dist, fix type errors

- packages/shared: tsconfig adds outDir/declaration, package.json adds build script
- apps/api/tsconfig.json: paths now point to shared/dist (.d.ts) instead of source,
  resolves TS6059 rootDir conflict
- apps/api/src/routes/{health,notifications,users}.ts: explicit Router type annotation
  to fix TS2742 portable type inference
- apps/api/src/events/handler.ts: cast via 'as unknown as' to satisfy TS2352
- apps/api/Dockerfile: build shared package before api

packages/shared/package.json

@@ -6,6 +6,7 @@
   "main": "./src/index.ts",
   "types": "./src/index.ts",
   "scripts": {
+    "build": "tsc",
     "test": "vitest run",
     "test:watch": "vitest"
   },