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/tsconfig.json

@@ -1,12 +1,15 @@
 {
   "compilerOptions": {
     "target": "ES2022",
-    "module": "ESNext",
-    "moduleResolution": "bundler",
+    "module": "NodeNext",
+    "moduleResolution": "NodeNext",
     "strict": true,
     "skipLibCheck": true,
     "declaration": true,
+    "declarationMap": true,
+    "rootDir": "./src",
     "outDir": "./dist"
   },
-  "include": ["src"]
+  "include": ["src"],
+  "exclude": ["node_modules", "dist", "**/*.test.ts"]
 }