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>

.gitignore

@@ -10,10 +10,11 @@ yarn-debug.log*
 yarn-error.log*
 pnpm-debug.log*
 
-# Build outputs
+# Build outputs (sync/dist is committed since server can't compile TypeScript)
 dist/
 build/
 out/
+!sync/dist/
 
 # Database
 *.sqlite