fix: commit pre-compiled dist and simplify sync Dockerfile

Server npm proxy silently drops devDependencies, making TypeScript
unavailable in Docker. Solution: compile locally and commit dist/.
Dockerfile now only needs prod deps + Playwright, both of which
install cleanly via the public registry.

Also fix TS2688/TS2304 errors: add DOM to tsconfig lib and cast
querySelectorAll results to Element inside $$eval callbacks.

Co-Authored-By: Claude Sonnet 4.6 <noreply@anthropic.com>

sync/tsconfig.json

@@ -2,7 +2,7 @@
   "compilerOptions": {
     "target": "ES2020",
     "module": "commonjs",
-    "lib": ["ES2020"],
+    "lib": ["ES2020", "DOM"],
     "outDir": "./dist",
     "rootDir": "./src",
     "strict": true,